A certain transverse sinusoidal wave of wavelength 20 cm is moving in the positive direction of

A certain transverse sinusoidal wave of wavelength 20 cm is...

A certain transverse sinusoidal wave of wavelength $20 \mathrm{~cm}$ is moving in the positive direction of an $x$ axis. The transverse velocity of the particle at $x=0$ as a function of time is shown in Fig. $16-49$, where the scale of the vertical axis is set by $u_{s}=5.0 \mathrm{~cm} / \mathrm{s}$. What are the (a) wave speed, (b) amplitude, and (c) frequency? (d) Sketch the wave between $x=0$ and $x=20 \mathrm{~cm}$ at $t=2.0 \mathrm{~s}$.

A certain transverse sinusoidal wave of wavelength $20 \mathrm{~cm}$ is moving in the positive direction of an $x$ axis. The transverse velocity of the particle at $x=0$ as a function of time is shown in
Fig. $16-49$, where the scale of the vertical axis is set by $u_{s}=5.0 \mathrm{~cm} / \mathrm{s}$. What are the (a) wave speed, (b) amplitude, and (c) frequency?
(d) Sketch the wave between $x=0$ and $x=20 \mathrm{~cm}$ at $t=2.0 \mathrm{~s}$.

Key Concepts

Phase and Phase Constants

The phase of a wave represents the fraction of a complete cycle reached at a specific point relative to a reference. Phase constants account for initial conditions or shifts in the wave and are important when relating the motion of the individual particles (through their velocities or displacements) to the overall wave pattern.

Graphical Representation of Waves

Sketching or graphing a wave provides a visual interpretation of the wave’s characteristics such as amplitude, wavelength, and phase. It involves plotting displacement against position at a fixed time, taking into account the wave’s sinusoidal nature and its propagation direction, which is essential for understanding spatial variations in the wave.

Wave Speed

Wave speed is the rate at which the wave propagates through a medium. It is determined by the product of the wavelength and frequency. Understanding wave speed is essential for predicting how fast disturbances travel along the medium, which links spatial and temporal characteristics of the wave.

Amplitude

Amplitude refers to the maximum displacement of a particle from its equilibrium position in a wave. It is an indicator of the energy carried by the wave and is directly related to the intensity of the wave's effect in the medium through which it propagates.

Wavelength

Wavelength is the spatial period of the wave—the distance over which the wave repeats its shape. It is a key parameter in defining the geometry and spatial structure of the wave and relates directly to other wave properties such as frequency and speed.

Transverse Sinusoidal Waves

These are waves in which the particle displacement is perpendicular to the direction of energy propagation and the waveform follows a sine or cosine pattern. The mathematical description of such waves involves sinusoidal functions that capture the periodic variation in displacement, making them foundational in studying oscillations and wave phenomena.

Frequency

Frequency is the number of complete oscillations or cycles that occur per unit time. Usually measured in hertz (Hz), frequency determines how rapidly the particles are oscillating. It is inversely related to the period of the wave and is a critical factor in understanding the temporal behavior of wave phenomena.

Transcript

00:01 Hi there, so for this problem we have certain transfer sinusoidal wave of wavelength that is equal to 20 centimeters.

00:13 And it is moving in the positive direction of the x -axis.

00:18 So the transfer velocity of the particle at x equals to zero as a function of time is shown in here.

00:31 So where the scale of the vertical axis is said to be u .s is equal to 5 centimeters per second.

00:45 So for part a of this problem, we are asked, what is the wave speed? so to calculate that, we know that the frequency is equal to 1 over the period.

01:06 In the period, as you can see from the figure, is simply four seconds.

01:16 So if we substitute that in here, we have one over four seconds.

01:21 That will give us one over four hertz.

01:27 And the speed is simply the product between the frequency and the wavelength.

01:34 And we know those values because we know that the frequency is 1 over 4 hearths and the wavelength is equal to 20 centimeters.

01:50 So from this we obtain a speed of 5 centimeters per second.

01:58 Now for part b we are asked about the amplitude and we refer to the graph to see that the maximum transverse speed, which we will refer to as um, is c.

02:17 Five centimeters five centimeters per second that is the amplitude for the function that we are given now the simply harmonic motion relation is that the maximum value for the speed is equal to the maximum value for the position times the angular frequency we know that the angular frequency is defined as 2 times pi times the frequency.

02:54 So we just simply substitute those values in here.

02:56 We will have that this is 5 is equal to the amplitude times 2 pi divided by 1 over 4.

03:08 And if we solve for the amplitude of the position, this will give us 3 .2 centimeters.

03:15 So that's a solution for part b of this problem.

03:19 Now for par c, we are asked about the frequency...

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