Equation (15.7) for a sinusoidal wave can be made more...

Equation (15.7) for a sinusoidal wave can be made more general by including a phase angle $\phi,$ where $0 \leq \phi \leq 2 \pi$ (in radians). Then the wave function $y(x, t)$ becomes $$ y(x, t)=A \cos (k x-\omega t+\phi) $$ (a) Sketch the wave as a function of $x$ at $t=0$ for $\phi=0$ , $\phi=\pi / 4, \phi=\pi / 2, \phi=3 \pi / 4,$ and $\phi=3 \pi / 2 .(\text { b) Calculate the }$ transverse velocity $v_{y}=\partial y / \partial t (\mathrm{c}) \mathrm{At} t=0,$ a particle on the string at $x=0$ has displacement $y=A / \sqrt{2}$ . Is this enough information to determine the value of $\phi ?$ In addition, if you are told that a particle at $x=0$ is moving toward $y=0$ at $t=0$ , what is the value of $\phi ?(\mathrm{d})$ Explain in general what you must know about the wave's behavior at a given instant to determine the value of $\phi$ .

Equation (15.7) for a sinusoidal wave can be made more general by including a phase angle $\phi,$ where $0 \leq \phi \leq 2 \pi$ (in radians). Then the wave function $y(x, t)$ becomes
$$
y(x, t)=A \cos (k x-\omega t+\phi)
$$
(a) Sketch the wave as a function of $x$ at $t=0$ for $\phi=0$ , $\phi=\pi / 4, \phi=\pi / 2, \phi=3 \pi / 4,$ and $\phi=3 \pi / 2 .(\text { b) Calculate the }$ transverse velocity $v_{y}=\partial y / \partial t
(\mathrm{c}) \mathrm{At} t=0,$ a particle on the string at $x=0$ has displacement $y=A / \sqrt{2}$ . Is this enough information to determine the value of $\phi ?$ In addition, if you are told that a particle at $x=0$ is moving toward $y=0$ at $t=0$ , what is the value of $\phi ?(\mathrm{d})$ Explain in general what you must know about the wave's behavior at a given instant to determine the value of $\phi$ .

Key Concepts

Initial Conditions in Wave Analysis

Initial conditions, such as the displacement and velocity of a particle on a wave at a specific time and position, are essential for determining unique solutions to wave functions. These conditions provide the necessary information to solve for unknown parameters, like the phase constant, and to predict the future behavior of the wave. Without adequate initial data, multiple wave configurations may be consistent with the observed state.

Sinusoidal Waves

Sinusoidal waves are periodic disturbances characterized by smooth, continuous oscillations. They can be described by functions such as cosine or sine, which capture the periodic behavior of physical waves such as sound waves, vibrations, and electromagnetic waves. The general form includes parameters like amplitude, frequency, and wavelength that determine the shape and behavior of the wave.

Phase Angle

The phase angle is a parameter added to the argument of the sinusoidal function that shifts the wave horizontally. It determines the starting point of the wave cycle relative to a reference position in space or time. Understanding the phase angle is crucial because it influences the initial conditions and the alignment between multiple waves, affecting phenomena such as interference and resonance.

Transverse Velocity

Transverse velocity refers to the rate of change of the wave’s displacement with respect to time, and it is obtained by differentiating the wave function with respect to time. In the context of waves on a string or similar media, it describes how fast a particular point on the medium moves up or down as the wave propagates.

Transcript

00:01 So here for this part we need to find how is the wave displaced when the phase angle changes.

00:11 So since the phase angle is changing from 0 to positive values and the positive values that are 3 times pi halves, which means 3 times 90 degrees.

00:22 So it's changing for 90 degrees with certain steps.

00:27 The figure here drawn and sketched shows the the position of the particle on the string along the x -axis and of course it shows the vertical displacement of the particle from the zero horizontal, which means the equilibrium position and then 0 .5 and negative 0 .5 of the amplitude.

00:56 So we see that the waveform, this is for, this first one is for the 0, then for the pi over 4, then for the pi over 2, then for the 3 pi over 4, and then 2 for the last one is for the 3 pi over 2.

01:22 This one is for in between these 2.

01:25 So we see that we have every waveform that is being drawn here is actually one the same waveform that is just moving to the right hand side as the phase grows from 0 to the final point of 3 pi over 2.

01:49 This is the only thing that will happen.

01:52 The waveform, starting and middle and ending position will just move right hand.

02:00 And it will move for one sixth, so we can basically derive the whole whole length on this sketch, horizontal length into every sixth part of it.

02:14 And each time the phase moves for one quarter of pi in the phase, the waveform will stay the same, but will move right hand to the right hand side for one sixth of the total x length given on this diagram here.

02:37 So this is how the wave will be displaced when the phase angle changes for one quarter of the pi angle in radiance.

02:51 Now for the part b, maybe it's better with the black color, so for the part b, we need to find this displacement and we know the displacement of the wave function.

03:12 So the wave function with displacement countenance, so the wave function with displacement counted in looks like x of t and this is amplitude cosine and we have all of the arguments kx minus omega t plus the phase shift displacement or the phase displacement only so then we should just find the transverse velocity which is partial derivative of y over t and notice that since the phase has nothing to do with time, and also k has nothing to do with time, only the negative omega will go in front.

03:59 So the derivative of cosine is sine and then the derivative of this element in the argument omega t is negative omega.

04:08 So it's just a omega sine of kx minus omega t plus phase.

04:19 And this is for the part b of this problem.

04:22 For the part c of this problem, we need to find from this given function of displacement, we need to substitute the given that the displacement is a over square root of 2...

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