{"id":11686,"date":"2024-11-28T14:19:35","date_gmt":"2024-11-28T14:19:35","guid":{"rendered":"https:\/\/fisicaevestibular.com.br\/novo\/?page_id=11686"},"modified":"2024-12-03T14:14:12","modified_gmt":"2024-12-03T14:14:12","slug":"equacao-da-onda-equacao-fundamental-da-ondulatoria-en","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/ondulatoria\/ondas\/equacao-da-onda-equacao-fundamental-da-ondulatoria-en\/","title":{"rendered":"Wave equation (Fundamental wave equation) – EN"},"content":{"rendered":"

Wave equation (Fundamental wave equation)<\/span><\/b><\/p>\n

Elements of a wave<\/span><\/b><\/p>\n

Consider the\u00a0<\/span><\/b>periodic wave (succession of equal pulses at equal times)\u00a0<\/span><\/b>in the following figures, whose\u00a0<\/span><\/b>main characteristics are:<\/span><\/b><\/p>\n

\"\"<\/p>\n

\"\"\u00a0\u00a0Wavelength (\u03bb\u00a0<\/span><\/b>)\u00a0<\/span><\/b>represents the distance traveled by the wave until it starts\u00a0<\/b>repeating again,\u00a0<\/b>that is, it is the\u00a0<\/b>shortest distance between two consecutive points\u00a0<\/b>that are in\u00a0<\/b>phase agreement,\u00a0<\/b>such as, for example, the\u00a0<\/b>shortest distance between two crests or two troughs.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b>\"\"<\/p>\n

\"\"\u00a0Period (T)\u00a0<\/span><\/b>time it takes\u00a0<\/b>for the\u00a0<\/b>wave\u00a0\u00a0<\/b>to\u00a0<\/b>travel one wavelength\u00a0<\/b>(1\u03bb),\u00a0<\/b>which is the\u00a0<\/b>same time it takes for any point on the wave\u00a0<\/b>to\u00a0<\/b>perform a\u00a0<\/b>complete oscillation\u00a0<\/b>(back and forth)\u00a0<\/b>and which\u00a0<\/b>is the same time it takes for the source to generate a complete wave.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b>\"\"<\/p>\n

In the\u00a0<\/span><\/b>figure above,\u00a0<\/span><\/b>the\u00a0<\/span><\/b>period\u00a0<\/span><\/b>T\u00a0<\/span><\/b>is the\u00a0<\/span><\/b>time between the instants T = t\u00a0<\/span><\/b>4<\/span><\/b><\/sub>\u00a0\u2013 t\u00a0<\/span><\/b>o<\/span><\/b><\/sub>\u00a0= t\u00a0<\/span><\/b>5<\/span><\/b><\/sub>\u00a0\u2013 t\u00a0<\/span><\/b>1<\/span><\/b><\/sub>\u00a0= t\u00a0<\/span><\/b>6<\/span><\/b><\/sub>\u00a0\u2013 t\u00a0<\/span><\/b>2<\/span><\/b><\/sub>\u00a0, etc.<\/span><\/b><\/p>\n

 <\/p>\n

\"\"\u00a0Frequency (f)\u00a0<\/span><\/b>Frequency\u00a0<\/b>(f)\u00a0<\/b>represents\u00a0<\/b>how many complete oscillations a wave makes.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/p>\n

in\u00a0<\/span><\/b>each unit of time,\u00a0<\/span><\/b>or\u00a0<\/span><\/b>how many wavelengths pass through a point on the wave in each unit of time.<\/span><\/b>\"\"<\/p>\n

If the\u00a0<\/span><\/b>period T is in seconds (s) the frequency will be in hertz (Hz),\u00a0<\/span><\/b>which means\u00a0<\/span><\/b>cycles per second.<\/span><\/b><\/p>\n

Example:\u00a0<\/span><\/b>If the\u00a0<\/span><\/b>frequency\u00a0<\/span><\/b>of a wave\u00a0<\/span><\/b>is 60 Hz\u00a0<\/span><\/b>, this means that the\u00a0<\/span><\/b>source or any point of this wave oscillates 60 times every second.<\/span><\/b><\/p>\n

Important relationship<\/span><\/b><\/p>\n

\"\"<\/p>\n

Wave propagation speed V (Fundamental wave equation)<\/span><\/b><\/p>\n

Assuming the\u00a0<\/span><\/b>medium is homogeneous,\u00a0<\/span><\/b>the\u00a0<\/span><\/b>wave\u00a0<\/span><\/b>propagates\u00a0<\/span><\/b>in it with constant speed\u00a0<\/span><\/b>, given by<\/span><\/b><\/p>\n

V =\u00a0\u00a0<\/span>\"\"S\/\u00a0<\/span>\"\"t.<\/span><\/b><\/p>\n

But note that\u00a0<\/span><\/b>when\u00a0\u00a0<\/span>\"\"S = 1\u03bb,\u00a0\u00a0<\/span>\"\"t = T\u00a0<\/span><\/b>, since the\u00a0<\/span><\/b>period T is the time it takes for the wave to travel one wavelength (1\u03bb).<\/span><\/b><\/p>\n

V =\u00a0\u00a0<\/span>\"\"S\/\u00a0<\/span>\"\"t\u00a0<\/span><\/b>V =\u00a0\u00a0<\/b>\u03bb<\/b><\/span>\u00a0\/T\u00a0\u00a0<\/b>\u00a0V =\u00a0\u00a0<\/b>\u03bb<\/b><\/span>\u00a0\/(1\/f)\u00a0\u00a0<\/b>V =\u00a0\u00a0<\/b>\u03bb<\/b><\/span>\u00a0.f (this equation is called the fundamental wave equation).<\/b><\/span>\u00a0\u00a0\"\"\u00a0\u00a0<\/b><\/b><\/b>\"\"<\/b><\/b><\/b>\"\"\u00a0\u00a0<\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

Pulse propagation speed in one-dimensional media (string)<\/span><\/b><\/p>\n

\u00a0\"\"<\/p>\n

Consider a\u00a0<\/span><\/b>pulse or several successive pulses (wave)\u00a0<\/span><\/b>propagating\u00a0<\/b><\/span>with\u00a0<\/span><\/b>speed\u00a0<\/b>V\u00a0<\/b>in a<\/b><\/span><\/b><\/b><\/b><\/b><\/p>\n

rope pulled\u00a0<\/span><\/b>(stretched) by a\u00a0<\/span><\/b>force of intensity F.<\/span><\/b><\/p>\n

We call\u00a0<\/span><\/b>the linear mass density (\u00b5)\u00a0<\/b>of a\u00a0<\/b>homogeneous string,\u00a0<\/b>with\u00a0<\/b>constant cross-section,\u00a0<\/b>which has\u00a0\u00a0<\/b>mass (m)\u00a0<\/b>\u00a0and\u00a0\u00a0<\/b>length (L),\u00a0<\/b>the expression:<\/b><\/span>\u00a0<\/b><\/b>\u00a0<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b>\u00a0<\/b><\/b><\/p>\n

The physical meaning of \u00b5\u00a0\u00a0<\/span>\"\"<\/b>means how much\u00a0<\/b>mass\u00a0<\/b>the string has per unit of length.\u00a0<\/b>Thus, a\u00a0<\/b>0.5 kg\/m string\u00a0<\/b>has, in SI,\u00a0<\/b>0.5 kg of mass for every 1 meter of length.<\/b><\/span>\u00a0<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b>\"\"<\/p>\n

 <\/p>\n

\"\"<\/p>\n

The\u00a0<\/span><\/b>speed (V) of propagation of the pulse in the string is\u00a0<\/span><\/b>also\u00a0<\/b><\/span>\u00a0given\u00a0<\/span><\/b>by:<\/b><\/span><\/b><\/b><\/b><\/p>\n

Where\u00a0<\/span><\/b>T is the intensity of the force\u00a0<\/span><\/b>pulling\u00a0<\/b><\/span>(\u00a0<\/span><\/b>stretching)\u00a0<\/b>the rope and \u00b5 is its linear mass density.<\/b><\/span><\/b><\/b><\/b><\/p>\n

Note that\u00a0<\/span><\/b>V is directly proportional to\u00a0\u00a0<\/span>\"\",\u00a0<\/span><\/b>that is,\u00a0<\/span><\/b>the more stretched the string is, the greater\u00a0<\/span><\/b>the\u00a0<\/span><\/b>propagation speed of the pulse or wave in it\u00a0<\/span><\/b>and that\u00a0<\/span><\/b>V\u00a0<\/span><\/b>is\u00a0<\/span><\/b>inversely proportional to\u00a0\u00a0<\/span>\"\",\u00a0<\/span><\/b>that is, the\u00a0<\/span><\/b>greater\u00a0<\/span><\/b>the\u00a0<\/span><\/b>linear density of the string, the lower\u00a0<\/span><\/b>the\u00a0<\/span><\/b>propagation speed of the wave in it.<\/span><\/b><\/p>\n

Wave propagation in two-dimensional media<\/span><\/b><\/p>\n

They can be of two types:<\/span><\/b><\/p>\n

\"\"<\/p>\n

Circular (spherical) waves\u00a0<\/span><\/b>When the\u00a0<\/b>tip of the ruler hits\u00a0<\/b>the\u00a0<\/b>surface of the water continuously and periodically,\u00a0<\/b>it creates\u00a0<\/b>circular disturbances\u00a0<\/b>(circular waves) that\u00a0<\/b>move on the surface of the water, moving away from the point where the disturbances are generated.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>wavefront or wave surface\u00a0<\/span><\/b>is the\u00a0<\/span><\/b>geometric locus\u00a0<\/span><\/b>of\u00a0<\/span><\/b>all points\u00a0<\/span><\/b>that are in\u00a0<\/span><\/b>agreement in vibration phase\u00a0<\/span><\/b>, such as\u00a0<\/span><\/b>two crests or two troughs.<\/span><\/b>\"\"<\/p>\n

The\u00a0<\/span><\/b>wave ray\u00a0<\/span><\/b>is a\u00a0<\/span><\/b>straight line perpendicular to the wave fronts\u00a0<\/span><\/b>and which indicates the\u00a0<\/span><\/b>direction and sense of propagation of these waves.<\/span><\/b><\/p>\n

The\u00a0<\/span><\/b>wavelength (\u03bb)\u00a0<\/span><\/b>is the\u00a0<\/span><\/b>distance between two consecutive wavefronts.<\/span><\/b><\/p>\n

\"\"<\/p>\n

 <\/p>\n

\"\"<\/p>\n

Plane (straight) waves<\/span><\/b>\u00a0<\/b><\/p>\n

\"\"<\/p>\n

Produced\u00a0<\/span><\/b>by gently and continuously touching the surface of the water with a ruler (fountain) in the position indicated in the figure.<\/span><\/b><\/p>\n

Note that the\u00a0<\/span><\/b>wave fronts and wave rays are parallel lines and therefore perpendicular.<\/span><\/b><\/p>\n

Wavelength\u00a0<\/span><\/b>(\u03bb)\u00a0<\/span><\/b>is the\u00a0<\/span><\/b>distance between two consecutive wavefronts (crests or troughs).<\/span><\/b><\/p>\n

 <\/p>\n

What you should know, information and tips<\/span><\/b><\/p>\n

\"\"<\/p>\n

Wave amplitude (A)<\/span><\/b><\/p>\n

\"\"<\/p>\n

Amplitude (A)\u00a0<\/span><\/b>corresponds to the\u00a0<\/b>distance of the ordinate (Y),\u00a0<\/b>in module,\u00a0<\/b>between the highest part\u00a0<\/b>(crest) or the\u00a0<\/b>lowest part\u00a0<\/b>(valley) and the\u00a0<\/b>midpoint (0).<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

Wavelength (\u03bb)<\/span><\/b><\/p>\n

\"\"<\/p>\n

It represents\u00a0<\/span><\/b>the\u00a0<\/span><\/b>distance traveled by the wave until it starts repeating again\u00a0<\/span><\/b>, that is, it is the\u00a0<\/span><\/b>shortest\u00a0<\/span><\/b>distance between two consecutive points\u00a0<\/b>that are in\u00a0<\/b>phase agreement\u00a0<\/b>, such as the\u00a0<\/b>shortest distance between two crests or two troughs.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

Period (T)\u00a0<\/span><\/b>time it takes for the wave to travel one wavelength (1\u03bb),\u00a0<\/b>which is the\u00a0<\/b>same time it takes for any point on the wave to oscillate\u00a0<\/b>(come and go)\u00a0<\/b>completely\u00a0<\/b>and which is the\u00a0<\/b>same time it takes for the source to generate a complete wave.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

Frequency (f)\u00a0<\/span><\/b>represents how many complete oscillations a wave makes in each unit of time,\u00a0<\/b>or\u00a0<\/b>how many complete wavelengths pass through a point on the wave in each unit of time.<\/b><\/span>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/p>\n

If the\u00a0<\/span><\/b>period T\u00a0<\/span><\/b>is\u00a0<\/span><\/b>in seconds (s) the frequency will be in hertz (Hz),\u00a0<\/span><\/b>which means\u00a0<\/span><\/b>cycles per second and the relationship between f and T\u00a0<\/span><\/b>is below:<\/span><\/b><\/p>\n

\"\"<\/p>\n

 <\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Formulas<\/span><\/b><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

If the\u00a0<\/span><\/b>medium is the same, V, \u03bb and f are\u00a0<\/span><\/b>also\u00a0<\/span><\/b>the same.<\/span><\/b><\/p>\n

\"\"<\/p>\n

The\u00a0<\/span><\/b>amplitude A is not related to V, f and \u03bb, but only to the amount of energy carried\u00a0<\/span><\/b>by the wave.\u00a0<\/span><\/b>The greater the energy, the greater the amplitude and vice versa.<\/span><\/b><\/p>\n

 <\/p>\n

\"\"<\/p>\n

Circular two-dimensional waves<\/span><\/b><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Two-dimensional plane waves<\/span><\/b><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Waves in three-dimensional media\u00a0<\/span><\/b>The wave\u00a0<\/span><\/b>propagates in space\u00a0<\/span><\/b>and in\u00a0<\/span><\/b>all directions.\u00a0<\/span><\/b>Example:\u00a0<\/span><\/b>Sound waves and light.<\/span><\/b><\/p>\n

\"\"<\/p>\n

The\u00a0<\/span><\/b>wavefronts are spheres and the wave rays are radial and leave the disturbance source in all directions.<\/span><\/b><\/p>\n

 <\/p>\n

\"\"\u00a0Interesting exercises you should check out:<\/span><\/b><\/p>\n

01-(FUVEST-SP)\u00a0<\/span><\/b>In a\u00a0<\/b>lake\u00a0<\/b>the\u00a0<\/b>wind produces periodic waves\u00a0<\/b>that\u00a0<\/b>propagate\u00a0<\/b>at the\u00a0<\/b>speed<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

of 2m\/s.\u00a0<\/span><\/b>The\u00a0<\/span><\/b>wavelength is 10m.\u00a0<\/span><\/b>Determine the\u00a0<\/span><\/b>period of oscillation of a boat:<\/span><\/b>\"\"<\/p>\n

a)\u00a0<\/span><\/b>when\u00a0<\/span><\/b>anchored in this lake.<\/span><\/b><\/p>\n

b)\u00a0<\/span><\/b>when it\u00a0<\/span><\/b>moves in the opposite direction to the propagation of the waves\u00a0<\/span><\/b>, at a\u00a0<\/span><\/b>speed of 8 m\/s.<\/span><\/b><\/p>\n

Resolution:<\/span><\/b><\/p>\n

a) With the boat anchored (stationary)\u00a0<\/span>\"\"V = 2 m\/s\u00a0<\/span>\"\"\u03bb = 10 m\u00a0<\/span>\"\"fundamental wave equation\u00a0<\/span>\"\"V = \u03bb.f\u00a0\u00a0<\/span>\"\"<\/b>\u00a0 2 = 10.f\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/b>f = 0.2 Hz\u00a0\u00a0<\/span>\"\"<\/b>T = 1\/f\u00a0\u00a0<\/b>\u00a0 T = 1\/0.2\u00a0\u00a0<\/b>T = 5 s.<\/b><\/span>\u00a0\u00a0<\/b>\"\"<\/b>\"\"\u00a0\u00a0<\/b><\/b><\/p>\n

b)\u00a0<\/span><\/b>As they\u00a0<\/span><\/b>move in opposite directions,\u00a0<\/span><\/b>the\u00a0<\/span><\/b>relative speed between the boat and the wave\u00a0<\/span><\/b>is\u00a0<\/span><\/b>V = 2 + 8\u00a0<\/span><\/b>V = 10 m\/s.<\/b><\/span>\u00a0\u00a0\"\"\u00a0<\/b><\/b>\"\"<\/p>\n

The wavelength remains the same \u03bb = 10 m = \u0394S\u00a0<\/span>\"\"so, to travel \u0394S = \u03bb = 10 m the wave must take the time of one period \u0394t = T with a speed of V = 10 m\/s\u00a0<\/span><\/b>V = \u0394S\/\u0394t\u00a0\u00a0<\/b><\/span>\u00a0\"\"\u00a0\u00a0<\/b>\"\"<\/b>\u00a0<\/b><\/p>\n

10 = 10\/T\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/b>T = 1 s.<\/span><\/b><\/p>\n

\"\"<\/p>\n

Using the\u00a0<\/span><\/b>given expression V =\u00a0<\/span>\"\":<\/span><\/b><\/p>\n

In the shallowest part\u00a0<\/span>\"\"V\u00a0<\/span><\/b>1<\/span><\/b><\/sub>\u00a0=\u00a0<\/span>\"\"\u00a0\"\"V\u00a0<\/span><\/b>1<\/span><\/b><\/sub>\u00a0= 5 m\/s.<\/span><\/b>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/b><\/p>\n

In the deepest part\u00a0\u00a0<\/span>\"\"<\/b>V\u00a0<\/b>2<\/b><\/sub>\u00a0V\u00a0<\/b>2<\/b><\/sub>\u00a0= 10 m\/s.<\/b><\/span>\u00a0\u00a0<\/b><\/b>\"\"\u00a0<\/b><\/sub>\u00a0\u00a0\"\"\u00a0<\/b><\/b><\/b><\/b><\/p>\n

Since, by the statement, the\u00a0<\/span><\/b>frequency f is the same\u00a0<\/span><\/b>V\u00a0<\/b>1<\/b><\/sub>\u00a0= \u03bb\u00a0<\/b>1<\/b><\/sub>\u00a0.f\u00a0\u00a0<\/b>\u00a0 5 = \u03bb\u00a0<\/b>1<\/b><\/sub>\u00a0.f\u00a0\u00a0<\/b>f = 5\/\u03bb\u00a0<\/b>1<\/b><\/sub>\u00a0(I)\u00a0<\/b>V\u00a0<\/b>2<\/b><\/sub>\u00a0= \u03bb\u00a0<\/b>2<\/b><\/sub>\u00a0.f\u00a0\u00a0<\/b>\u00a0 10 = \u03bb\u00a0<\/b>2<\/b><\/sub>\u00a0.f\u00a0\u00a0<\/b>f = 10\/\u03bb\u00a0<\/b>2<\/b><\/sub>\u00a0(II).<\/b><\/span>\u00a0\u00a0\"\"\u00a0<\/b><\/b><\/b><\/b><\/b>\"\"<\/b><\/b><\/b>\"\"\u00a0<\/b><\/b><\/b>\"\"<\/b><\/b><\/b><\/b>\"\"<\/b><\/b><\/b>\"\"<\/b><\/b><\/b><\/b><\/p>\n

Equating (I) with (II)\u00a0<\/span>\"\"\u00a0<\/b>5\/\u00a0<\/span><\/b>\u03bb\u00a0<\/span><\/b><\/span>1<\/span><\/b><\/sub>\u00a0= 10\/\u00a0<\/span><\/b>\u03bb\u00a0<\/span><\/b><\/span>2\u00a0<\/span><\/b><\/sub>\u00a05\u00a0<\/b>\u03bb\u00a0<\/b>2<\/b><\/sub>\u00a0= 10\u03bb\u00a0<\/b>1<\/b><\/sub>\u00a0\u03bb\u00a0<\/b>2<\/b><\/sub>\u00a0= 2\u03bb\u00a0<\/b>1<\/b><\/sub>\u00a0.<\/b><\/span>\u00a0\u00a0\"\"\u00a0\u00a0<\/b><\/b><\/b><\/b><\/b><\/b>\u00a0\"\"<\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

R \u2013 B\u00a0<\/span><\/b><\/p>\n

03-(UNICAMP-SP)\u00a0<\/span><\/b>The\u00a0<\/b>GPS system (\u201cGlobal Positioning System\u201d)\u00a0<\/b>consists of a\u00a0<\/b>set of satellites in orbit around the Earth\u00a0<\/b>that\u00a0<\/b>transmit electromagnetic signals\u00a0<\/b>to\u00a0<\/b>receivers\u00a0<\/b>on the\u00a0<\/b>Earth’s surface.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>propagation speed\u00a0<\/span><\/b>of the signals is\u00a0<\/span><\/b>300,000 km\/s.\u00a0<\/span><\/b>For the\u00a0<\/span><\/b>system to work well\u00a0<\/span><\/b>,\u00a0<\/span><\/b>atmospheric absorption of this electromagnetic signal must be small\u00a0<\/span><\/b>.<\/span><\/b><\/p>\n

The\u00a0<\/span><\/b>following figure\u00a0<\/span><\/b>shows the\u00a0<\/span><\/b>percentage of electromagnetic radiation absorbed by the atmosphere as a function of wavelength.<\/span><\/b><\/p>\n

\"\"<\/p>\n

a)\u00a0<\/span><\/b>The\u00a0<\/span><\/b>frequency of the GPS signal\u00a0<\/span><\/b>is\u00a0<\/span><\/b>1,500 MHz.\u00a0<\/span><\/b>What is the\u00a0<\/span><\/b>corresponding wavelength?\u00a0<\/span><\/b>What\u00a0<\/span><\/b>percentage of the signal is absorbed by the atmosphere?<\/span><\/b><\/p>\n

b)\u00a0<\/span><\/b>One of the\u00a0<\/span><\/b>most important applications of the GPS system\u00a0<\/span><\/b>is\u00a0<\/span><\/b>determining the position\u00a0<\/span><\/b>of a certain\u00a0<\/span><\/b>receiver on Earth.<\/span><\/b><\/p>\n

This\u00a0<\/span><\/b>determination\u00a0<\/span><\/b>is\u00a0<\/span><\/b>made by measuring the time\u00a0<\/span><\/b>it takes for the signal\u00a0<\/span><\/b>to travel from the satellite to\u00a0<\/span><\/b>the\u00a0<\/span><\/b>receiver.<\/span><\/b><\/b><\/p>\n

What is the\u00a0<\/span><\/b>variation \u0394t in the time measurement\u00a0<\/span><\/b>made by the\u00a0<\/span><\/b>receiver\u00a0<\/span><\/b>that\u00a0<\/span><\/b>corresponds to a variation\u00a0<\/span><\/b>in the\u00a0<\/span><\/b>satellite-receiver distance of \u0394x = 100 m?\u00a0<\/span><\/b>Assume that the\u00a0<\/span><\/b>signal trajectory is rectilinear.<\/span><\/b><\/p>\n

Resolution:<\/span><\/b><\/p>\n

a)\u00a0<\/span><\/b>Given\u00a0<\/span>\"\"<\/b>V = 300,000 km\/s = 3.10\u00a0<\/span><\/b>8<\/span><\/b><\/sup>\u00a0m\/sef = 1,500MHz = 1500.10\u00a0<\/span><\/b>6<\/span><\/b><\/sup>\u00a0Hz.<\/span><\/b><\/p>\n

Applying the\u00a0<\/span><\/b>fundamental wave equation V = \u03bbf\u00a0<\/span>\"\"<\/b>\u00a0\u00a0 3.10\u00a0<\/span><\/b>8<\/span><\/b><\/sup>\u00a0= \u03bb.1,500.10\u00a0<\/span><\/b>6<\/span><\/b><\/sup>\u00a0\u03bb\u00a0<\/b>= 2.0.10\u00a0<\/b>-1<\/b><\/sup>\u00a0m.<\/b><\/span>\u00a0\"\"\u00a0\u00a0<\/b><\/b><\/b><\/b><\/b><\/p>\n

See from\u00a0<\/span><\/b>the graph\u00a0<\/span><\/b>that for this\u00a0<\/span><\/b>wavelength\u00a0<\/span><\/b>(\u00a0<\/span><\/b>\u03bb\u00a0<\/span><\/b>= 2.0.10\u00a0<\/span><\/b>-1<\/span><\/b><\/sup>\u00a0m)\u00a0<\/span><\/b>,\u00a0<\/span><\/b>the\u00a0<\/span><\/b>percentage (fraction) of signal absorption by the atmosphere is zero.<\/span><\/b><\/p>\n

b)\u00a0\u00a0<\/span><\/b><\/span>\u0394\u00a0<\/span><\/b>x =<\/span><\/b><\/span>\u00a0\u0394\u00a0<\/span><\/b>S =100m\u00a0<\/span>\"\"V =<\/span><\/b><\/span>\u00a0\u0394\u00a0<\/span><\/b>S\/<\/span><\/b><\/span>\u00a0\u0394\u00a0<\/span><\/b>t\u00a0\u00a0<\/span><\/b><\/span>\u00a0 3.10\u00a0<\/b><\/span>8<\/b><\/span><\/sup>\u00a0= 100\/<\/b><\/span>\u00a0\u0394\u00a0<\/b>t\u00a0\u00a0<\/b><\/span>\u0394\u00a0<\/b>t = 3.3.10\u00a0<\/b><\/span>-7<\/b><\/span><\/sup>\u00a0\u00a0s<\/b><\/span><\/span>\u00a0\"\"<\/b><\/span><\/b><\/b><\/b>\"\"\u00a0\u00a0<\/b><\/span><\/b><\/b><\/b><\/b><\/p>\n

04-(FUVEST-SP)\u00a0<\/span><\/b>A sensor, mounted on a Petrobr\u00e1s platform, with a fixed position in relation to the seabed, records the successive positions of a small ball floating\u00a0<\/b>on\u00a0<\/b>the\u00a0<\/b>surface\u00a0<\/b>of\u00a0<\/b>the\u00a0<\/b>water\u00a0<\/b>,\u00a0<\/b>as\u00a0<\/b>a\u00a0<\/b>sea\u00a0<\/b>wave\u00a0<\/b>passes\u00a0<\/b>over\u00a0<\/b>this\u00a0<\/b>ball continuously.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>ball describes an approximately circular motion\u00a0<\/span><\/b>in the\u00a0<\/span><\/b>vertical plane,\u00a0<\/span><\/b>remaining around the\u00a0<\/span><\/b>same mean position\u00a0<\/span><\/b>, as reproduced in the\u00a0<\/span><\/b>sequence of recordings below\u00a0<\/span><\/b>, at the\u00a0<\/span><\/b>times indicated.\u00a0<\/span><\/b>The\u00a0<\/span><\/b>interval between recordings is shorter than the wave period.<\/span><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>propagation speed\u00a0<\/span><\/b>of this\u00a0<\/span><\/b>sinusoidal wave is 1.5 m\/s.<\/span><\/b><\/p>\n

For\u00a0<\/span><\/b>these conditions:<\/span><\/b>\"\"<\/p>\n

a)\u00a0<\/span><\/b>Determine the\u00a0<\/b>period T,\u00a0<\/b>in\u00a0<\/b>seconds,\u00a0<\/b>of this\u00a0<\/b>sea wave.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

b)\u00a0<\/span><\/b>Determine the\u00a0<\/b>wavelength\u00a0<\/b>,\u00a0<\/b>in\u00a0<\/b>m,\u00a0<\/b>of this\u00a0<\/b>sea wave.<\/b><\/span>\u00a0<\/b><\/b>\"\"<\/b><\/b><\/b><\/b><\/b><\/p>\n

c)\u00a0<\/span><\/b>Draw a\u00a0<\/b>diagram of the profile of this wave\u00a0<\/b>at\u00a0<\/b>time t = 14 s,\u00a0<\/b>as\u00a0<\/b>seen from the fixed platform.\u00a0<\/b>Indicate the\u00a0<\/b>appropriate values \u200b\u200bon the horizontal and vertical axes.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

Resolution:<\/span><\/b>\u00a0<\/b><\/p>\n

a)\u00a0<\/span><\/b>It is a\u00a0<\/b>mixed wave,\u00a0<\/b>as\u00a0<\/b>it oscillates horizontally and vertically\u00a0<\/b>and these\u00a0<\/b>oscillations\u00a0<\/b>in a\u00a0<\/b>mixed wave occur at the same time in these two directions.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

Note\u00a0<\/span><\/b>in the figures above\u00a0<\/span><\/b>that the\u00a0<\/span><\/b>time interval of 6s\u00a0<\/span><\/b>corresponds to\u00a0<\/span><\/b>three quarters of the period T,\u00a0<\/span><\/b>that is,\u00a0<\/span><\/b>T=8s.<\/span><\/b><\/p>\n

b)\u00a0<\/span><\/b>f=1\/T\u00a0\u00a0<\/span>\"\"<\/b>f = (1\/8) Hz\u00a0\u00a0<\/b>V = \u03bb.f\u00a0\u00a0<\/b>\u00a0 1.5 = \u03bb.(1\/8) \u00a0\u00a0<\/b>\u03bb = 12 m.<\/b><\/span>\u00a0\u00a0<\/b>\"\"<\/b>\u00a0\u00a0<\/b>\"\"<\/b>\"\"\u00a0\u00a0<\/b><\/b><\/p>\n

w)<\/span><\/b><\/span><\/p>\n

\"\"<\/p>\n

05-(UFRJ-RJ)\u00a0<\/span><\/b>Through a\u00a0<\/b>suitable device , waves\u00a0<\/b>are produced\u00a0<\/b>in an elastic medium,\u00a0<\/b>in such a way that the\u00a0<\/b>frequencies of the waves obtained are in the range of 15Hz to 60Hz.\u00a0<\/b>The graph shows how the wavelength (\u03bb) varies as a function of the frequency (f).<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

a)\u00a0<\/span><\/b>Calculate the\u00a0<\/span><\/b>shortest wavelength produced in this experiment.<\/span><\/b><\/p>\n

b)\u00a0<\/span><\/b>For a\u00a0<\/b>wavelength of 12 m, calculate the period of the wave.<\/b><\/span><\/b><\/b><\/b><\/p>\n

Resolution:<\/span><\/b>\u00a0<\/b><\/p>\n

a)\u00a0<\/span><\/b>When\u00a0<\/span><\/b>f = 30 Hz, \u03bb = 12 m\u00a0<\/span>\"\"fundamental wave equation\u00a0<\/span>\"\"V =\u00a0<\/span>\"\".f\u00a0\u00a0<\/span>\"\"<\/b>V = 12.30\u00a0\u00a0<\/b><\/span>\u00a0\u00a0<\/b>\"\"<\/b><\/p>\n

V = 360 m\/s\u00a0<\/span><\/b>(constant-same medium).<\/b><\/span><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>smallest \u03bb\u00a0<\/span><\/b>\u00a0occurs when\u00a0<\/span><\/b>f = 60 Hz (see graph)\u00a0\u00a0<\/span>\"\"<\/b>\u00a0 360 = \u03bb.60\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/b>\u03bb\u00a0<\/span><\/b>minimum<\/span><\/b><\/sub>\u00a0= 6 m.<\/span><\/b><\/p>\n

b)\u00a0<\/span><\/b>When\u00a0\u00a0<\/b>\u03bb = 12 m, f = 30 Hz\u00a0\u00a0<\/b>T = 1\/f\u00a0\u00a0<\/b>T = (1\/30) s.<\/b><\/span>\u00a0<\/b><\/b>\"\"<\/b>\u00a0\u00a0<\/b>\"\"<\/b>\u00a0\u00a0<\/b><\/b><\/p>\n

06-(FUVEST-SP)\u00a0<\/span><\/b>A\u00a0<\/b>large aquarium,\u00a0<\/b>with\u00a0<\/b>glass side walls,\u00a0<\/b>allows\u00a0<\/b>you to see\u00a0<\/b>a wave propagating\u00a0<\/b>on the surface of the water.<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

The\u00a0<\/span><\/b>figure\u00a0<\/span><\/b>represents the\u00a0<\/span><\/b>profile of such a wave at instant T\u00a0<\/span><\/b>o<\/span><\/b><\/sub>\u00a0. During\u00a0<\/span><\/b>its passage,\u00a0<\/span><\/b>a\u00a0<\/span><\/b>buoy,\u00a0<\/span><\/b>in a given\u00a0<\/span><\/b>position,\u00a0<\/span><\/b>oscillates up and down\u00a0<\/b>and its\u00a0<\/b>vertical displacement (y),\u00a0<\/b>as\u00a0<\/b>a function\u00a0<\/b>of\u00a0<\/b>time,\u00a0<\/b>is represented in the graph<\/b><\/span><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/b><\/p>\n

\"\"<\/p>\n

<\/b>Using this\u00a0<\/span><\/b>information,\u00a0<\/span><\/b>calculate\u00a0<\/b><\/span>the\u00a0<\/span><\/b>wave propagation speed.<\/span><\/b><\/p>\n

Resolution:<\/span><\/b><\/b><\/p>\n

From the figure, the\u00a0<\/span><\/b>wavelength \u03bb\u00a0<\/span><\/b>can be the\u00a0<\/span><\/b>distance between two successive crests\u00a0\u00a0<\/span>\"\"\u03bb = 20 m.<\/span><\/b><\/p>\n

From the graph, the\u00a0<\/span><\/b>period T is the time interval that the wave takes to start repeating\u00a0\u00a0<\/span>\"\"T =<\/span><\/b><\/p>\n

10 s\u00a0<\/span>\"\"\u00a0<\/b>f = 1\/T\u00a0\u00a0<\/span>\"\"\u00a0<\/b>f = (1\/10) Hz.<\/span><\/b><\/p>\n

Fundamental wave equation\u00a0\u00a0<\/span>\"\"V = \u03bb.f\u00a0\u00a0\u00a0<\/span>\"\"<\/b>\u00a0\u00a0V = 20.(1\/10)\u00a0\u00a0<\/span>\"\"\u00a0\u00a0<\/b>V = 2.0 m\/s.<\/span><\/b><\/p>\n

Check out the exercises with commented solutions<\/span><\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"

Wave equation (Fundamental wave equation) Elements of a wave Consider the\u00a0periodic wave (succession of equal pulses at equal times)\u00a0in the following figures, whose\u00a0main characteristics are: \u00a0\u00a0Wavelength (\u03bb\u00a0)\u00a0represents the distance traveled by the wave until it starts\u00a0repeating again,\u00a0that is, it is the\u00a0shortest distance between two consecutive points\u00a0that are in\u00a0phase agreement,\u00a0such as, for example, the\u00a0shortest distance between two crests or two troughs.\u00a0<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2398,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-11686","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/11686","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/comments?post=11686"}],"version-history":[{"count":2,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/11686\/revisions"}],"predecessor-version":[{"id":12064,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/11686\/revisions\/12064"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2398"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=11686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}