A water wave traveling in a straight line on a lake is...

A water wave traveling in a straight line on a lake is described by the equation $$ y(x, t)=(2.75 \mathrm{~cm}) \cos (0.410 \mathrm{rad} / \mathrm{cm} x+6.20 \mathrm{rad} / \mathrm{s} t) $$ where $y$ is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

A water wave traveling in a straight line on a lake is described by the equation
$$
y(x, t)=(2.75 \mathrm{~cm}) \cos (0.410 \mathrm{rad} / \mathrm{cm} x+6.20 \mathrm{rad} / \mathrm{s} t)
$$
where $y$ is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

Key Concepts

Wave Equation

The wave equation mathematically describes the displacement of a point on a wave as a function of both space and time. It encapsulates the behavior of the wave, including its amplitude, wavelength, period, and phase, and serves as the starting point for analyzing various physical characteristics of the wave, such as speed and oscillation frequency.

Angular Frequency (?)

Angular frequency represents the rate at which the wave oscillates in time and is expressed in radians per second. It is directly related to the frequency of the wave by the relation ? = 2?f and is crucial for determining the time period and dynamic behavior of the wave.

Wave Number (k)

The wave number is a measure of the number of radians per unit distance along the direction of wave propagation. It is linked to the wavelength by the relation k = 2?/? and provides insight into the spatial periodicity of the wave.

Wave Period

The wave period is the time required for the wave to complete one cycle of its oscillation. It is inversely related to the frequency and can be calculated using T = 2?/?. This concept is essential for understanding how long a particular phase of the wave lasts at a fixed point in space.

Wavelength

Wavelength is the spatial period of the wave—the distance over which the wave's shape repeats. It is inversely related to the wave number by the relation ? = 2?/k and is key to determining the spatial distribution of the wave's energy.

Phase Velocity

Phase velocity is the rate at which a fixed phase of the wave (for example, the crest) travels through space. It is found by taking the ratio of the angular frequency to the wave number, v = ?/k, and provides insight into how fast details of the wave pattern propagate.

Maximum Particle Speed

Maximum particle speed refers to the highest speed attained by a point on the medium (such as the cork or bobber) due to the wave's oscillation. This speed is typically calculated as the product of the amplitude and the angular frequency, indicating the maximum velocity of vertical motion induced by the wave.

Transcript

00:01 So we should first write the equation that describes these waves, these waves.

00:05 And this is going to be y of xt.

00:08 It's going to be 0 .0275 meters.

00:14 This is 2 .75 centimeters.

00:16 However, i just wanted to convert into meters immediately.

00:20 So we'll have cosine, and then it'll be times cosine.

00:24 And then this will be 0 .410 radians per centimeter.

00:30 Times x and then plus 6 .20 radiance per second times t.

00:41 So this will be our equation for the wave.

00:44 And then we know that a standard equation for a wave is going to look like this.

00:52 A cosine of k x minus omega t.

00:58 And then we are asked, the question is asking us, how long will it take for a waveform essentially for a complete waveform to pass by the fishermen? and the time to complete one wave pattern is called the period.

01:15 So we can say that omega is going to be equal to 2 pi over t, where t is the period.

01:22 T is going to be the time for one complete wave pattern to pass the fisherman.

01:27 We simply need to solve for t.

01:31 And this is going to be 2 pi divided by 6 .2 radiance per second.

01:37 We found the omega here by relating this equation to this equation.

01:45 And the coefficient of the t is going to line up with the omega or the angular frequency of this wave.

01:52 So the angular frequency of the wave is found simply by taking it from the original equation.

01:58 And this is going to be equal to 1 .013 seconds.

02:04 Now, they're asking us to find the horizontal distance covered so we can use the wave number.

02:12 This is going to be 2 pi over lambda.

02:15 And the horizontal distance covered in one complete wave pattern would be considered the wavelength of the wave.

02:22 So it's convenient that we can just use the equation for the wave number, isolate the wavelength, we can plug in for our constants.

02:32 K, we're gunning it from the equation.

02:35 It's going to be this variable right here.

02:37 So 0 .410 radians per centimeter.

02:42 And then of course the wavelength is going to be 15 .325 centimeters.

02:50 Or we can say the wavelength is equal to 0 .15325 meters.

03:01 And then at this point, we can find the frequency.

03:05 So part b is asking us to find the frequency...

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