A sinusoidal transverse wave of wavelength 20 cm travels along a string in the positive directio

A sinusoidal transverse wave of wavelength 20 cm travels...

A sinusoidal transverse wave of wavelength 20 $\mathrm{cm}$ travels along a string in the positive direction of an $x$ axis. The displacement $y$ of the string particle at $x=0$ is given in Fig. $16-34$ as a function of time $t .$ The scale of the vertical axis is set by $y_{s}=4.0 \mathrm{cm} .$ The wave equation is to be in the form $y(x, t)=y_{m} \sin (k x \pm \omega t+\phi) \cdot(a)$ At $t=0,$ is a plot of $y$ versus $x$ in the shape of a positive sine function or a negative sine function? What are (b) $y_{m},(\mathrm{c}) k,(\mathrm{d}) \omega,(\mathrm{e}) \phi,(\mathrm{f})$ the sign in front of $\omega,$ and $(\mathrm{g})$ the speed of the wave? (h) What is the transverse velocity of the particle at $x=0$ when $t=5.0 \mathrm{s} ?$

A sinusoidal transverse wave of wavelength 20 $\mathrm{cm}$ travels along a
string in the positive direction of an
$x$ axis. The displacement $y$ of the
string particle at $x=0$ is given in Fig. $16-34$ as a function of time $t .$ The scale of the vertical axis is
set by $y_{s}=4.0 \mathrm{cm} .$ The wave equation is to be in the form
$y(x, t)=y_{m} \sin (k x \pm \omega t+\phi) \cdot(a)$ At $t=0,$ is a plot of $y$ versus $x$ in
the shape of a positive sine function or a negative sine function? What are (b) $y_{m},(\mathrm{c}) k,(\mathrm{d}) \omega,(\mathrm{e}) \phi,(\mathrm{f})$ the sign in front of $\omega,$ and $(\mathrm{g})$ the speed of the wave? (h) What is the transverse velocity of the particle at $x=0$ when $t=5.0 \mathrm{s} ?$

Key Concepts

Direction of Wave Propagation

The sign in front of ? in the wave equation indicates the direction in which the wave is traveling. A negative sign is typically used for waves traveling in the positive x-direction, as it ensures that the phase remains constant for points moving with the wave.

Transverse Velocity

The transverse velocity is the rate of change of the displacement of a medium particle in the direction perpendicular to the motion of the wave. It is obtained by taking the partial derivative of the wave function with respect to time, reflecting how fast a point on the string moves up or down as the wave passes through.

Wave Speed

The wave speed (v) is the rate at which the wave propagates through the medium. It can be calculated using the relationship v = ?/k or v = f?, linking the temporal and spatial characteristics of the wave.

Phase Constant

The phase constant (?) sets the initial condition for the wave at time t = 0 and position x = 0. It determines the starting point on the sine (or cosine) curve and is crucial for accurately modeling the wave’s behavior at a given reference time and position.

Amplitude

The amplitude (y_m) is the maximum displacement of the wave from its equilibrium position. It represents the energy carried by the wave and is determined by the scale factor of the sinusoidal function in the wave equation.

Wavelength and Wave Number

The wavelength (?) is the spatial period of the wave, which is the distance over which the wave's shape repeats. The wave number (k) is related to the wavelength by the relation k = 2?/?. It quantifies how many radians the phase changes per unit distance.

Angular Frequency

Angular frequency (?) describes how rapidly the wave oscillates in time, measured in radians per second. It is connected to the frequency (f) of the wave by the formula ? = 2?f, indicating how many cycles occur in one second.

Sinusoidal Wave

A sinusoidal wave is a periodic disturbance that can be described using sine or cosine functions. Its shape is smooth and repetitive, making it ideal for modeling many natural phenomena such as vibrations and light waves.

Wave Equation

The wave equation in the form y(x, t) = y_m sin(k x ± ? t + ?) encapsulates all the main parameters that characterize a traveling wave. It shows how the displacement y at position x and time t depends on the amplitude (y_m), the wave number (k), the angular frequency (?), and a phase shift (?), with the sign before ? t determining the direction of propagation.

Transcript

00:01 So in this question, we are given this graph, and this graph, unlike the y versus x graph, we usually see, is a y versus t graph.

00:11 And then what it says is that this is a graph of 1 point at x equals 0 of the sinusoid wave, as it travels over time.

00:23 So let's look at the questions.

00:28 We are given a couple numbers.

00:29 We know it takes a number.

00:30 We know it takes, 10 seconds to travel a whole period.

00:34 We know that maximum minimum amplitude is 4 .0 centimeter, and we know the wave of the wave is 20 centimeter.

00:43 Question a asks, if we draw the y versus x graph, is this graph going to be a positive sign or a negative sign? well, let's look at this.

00:55 Right now we have y versus t graph.

00:58 If we draw a y versus x graph, at first, this point stays at zero, and as times goes by, it goes to positive, and then it go back to normal.

01:14 Well, let's try out what would happen if we have a positive sine wave.

01:19 If we have a positive sine wave, and we know this wave is, let's see, do we know where this wave is traveling? yes, we know this wave.

01:34 Is traveling to the right.

01:39 So if this wave is traveling to the right, what this point would do is that it has this form of this, and then as this wave traveled to the right, you can imagine, well, now it travels to the right, and this point is going to go down, so that is not correct.

01:56 As opposed to, it's a negative sine wave, and then it travels to the right, this point is going to go up.

02:11 And eventually it goes up, it goes down, it goes up, and then it goes to the minimal, and it goes up, which is just as, just like this graph.

02:21 This graph is the position of x, and it goes up, it goes down, it goes to minimal, and then it goes to equilibrium.

02:30 So from this, we can see that a negative side, y -x graph, will best correspond to this y -t graph of this point.

02:45 So i hope that makes sense because yt graph and yx graph are two different types of graphs and even though they look similar, so we present different things and you need to learn to how to respond one to the other okay, so for part b we need to find one max and this is easy because the graph already told us maximum one is four centimeters, so we know this and then we need to find k k is 2 pi over lambda so question told us lambda is 20 centimeter so this is pi over 10, which is 0 .3 .1 radius per centimeter.

03:28 Then we also need to figure out what omega is.

03:31 In order to find omega, we need to figure out.

03:38 So different ways we can find omega, we can find it by using k times v.

03:43 Because then we need to find out v.

03:45 So i find the easiest way to just use a period because we know period is 10 second.

03:50 So omega corresponds to frequency by 2 pi f, and f is 1 over t, so omega is just 2 pi of a period...

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