A sinusoidal transverse wave is traveling along a string in...

A sinusoidal transverse wave is traveling along a string in the negative direction of an $x$ axis. Figure $16-35$ shows a plot of the displacement as a function of position at time $t=0 ;$ the scale of the $y$ axis is set by $y_{s}=4.0 \mathrm{cm} .$ The string tension is $3.6 \mathrm{N},$ and its linear density is 25 $\mathrm{g} / \mathrm{m} .$ Find the (a) amplitude, (b) wavelength, (c) wave speed, and (d) period of the wave. (e) Find the maximum transverse speed of a particle in the string. If the wave is of the form $y(x, t)=y_{m} \sin (k x \pm \omega t+\phi),$ what are $(\mathrm{f}) k,(\mathrm{g}) \omega,(\mathrm{h})$ $\phi,$ and (i) the correct choice of sign in front of $\omega ?$

A sinusoidal transverse wave is traveling along a string in
the negative direction of an $x$ axis.
Figure $16-35$ shows a plot of the displacement as a function of position at time $t=0 ;$ the scale of the
$y$ axis is set by $y_{s}=4.0 \mathrm{cm} .$ The string tension is $3.6 \mathrm{N},$ and its linear density is 25 $\mathrm{g} / \mathrm{m} .$ Find the (a) amplitude, (b) wavelength,
(c) wave speed, and (d) period of the wave. (e) Find the maximum transverse speed of a particle in the string. If the wave is of
the form $y(x, t)=y_{m} \sin (k x \pm \omega t+\phi),$ what are $(\mathrm{f}) k,(\mathrm{g}) \omega,(\mathrm{h})$
$\phi,$ and (i) the correct choice of sign
in front of $\omega ?$

Key Concepts

Sinusoidal Transverse Wave

This concept refers to a wave that exhibits a sinusoidal (sine or cosine) variation in displacement as a function of both position and time, where the displacement is perpendicular to the direction of propagation. The general mathematical form is y(x, t) = y? sin(kx ± ?t + ?), which provides a compact description of its spatial and temporal variations.

Wave Propagation Direction

The sign in front of the ?t term in the wave equation (i.e., the ± sign) is used to indicate the direction in which the wave travels along the x-axis. The choice of sign correlates with whether the wave is propagating in the positive or negative direction.

Wave Number (k)

The wave number represents the number of wave cycles per unit distance and is defined as k = 2?/?. It characterizes the spatial frequency of a wave and plays a crucial role in describing the wave's oscillatory behavior in space.

Maximum Transverse Speed

This is the peak speed of a particle in the medium moving perpendicular to the direction of wave propagation. It occurs when the particle's displacement is increasing or decreasing most rapidly, and is typically calculated as the product of the angular frequency and the amplitude (?y?).

Phase Constant (?)

The phase constant determines the initial phase of the wave at time zero and position zero. It shifts the wave along the time or space axis, ensuring that the mathematical description aligns correctly with the physical initial conditions.

Wave Speed

Wave speed is the rate at which the wave propagates through the medium. In many physical scenarios, including waves on a string, this speed is determined by properties of the medium—in the case of a string, by the tension and mass per unit length.

Wavelength

Wavelength is the spatial period of the wave; it is the distance over which the wave's shape repeats itself. It is directly related to the wave number via the relation ? = 2?/k, making it a fundamental characteristic of the wave's structure.

Amplitude

Amplitude is the maximum displacement from the equilibrium position a point on the medium reaches during oscillation. It is a key parameter that indicates the energy carried by the wave, as larger amplitudes generally correspond to higher energy transmission.

Period

The period is the time taken for one complete oscillation or cycle of the wave at a fixed position. It is the reciprocal of the frequency and is directly related to the angular frequency by T = 2?/?.

Angular Frequency (?)

Angular frequency describes how rapidly the wave oscillates in time and is given by ? = 2?f, where f is the frequency. It is a fundamental parameter in linking the temporal behavior of the wave to its spatial characteristics through the wave equation.

Transcript

00:01 So in this question, we are given this graph, and then from this graph, we have to figure out multiple properties of this wave.

00:09 So let's look at this graph.

00:11 A couple of things we need to read is when different look at the amplitude of it.

00:20 So from the graph, ys is 4 centimeter, but the amplitude and ys has 4 ticks.

00:26 It looks like the amplitude has 5 ticks, so this is 5 .0 centimeters.

00:33 Then we can look at lambda.

00:36 It looks like from this graph that from 0 to 40 is about one period.

00:41 So we will say this is 0 .40 meter.

00:46 And if you are in doubt, you can also go more precisely.

00:50 You can see that we have a peak around 5 cm, and then we have a peak roughly at 45 centimeters.

00:57 So if you subtract them, they are also around 40 centimeters.

01:02 And then we have speed.

01:03 In order to look for the speed, we are given the tension and the mass density, and that is enough for us to calculate the speed, because the speed is tension, 3 .6 md, of the density, which is 25 times 10 to negative 3 kilograms per meter.

01:21 That gives us 12 meter per second.

01:25 Then period.

01:26 Well, we can find period if we already know velocity and wavelength, because period equals 1 over frequency, and a frequent.

01:35 Is velocity over lambda.

01:38 So this is lambda over velocity.

01:47 Lambda is 0 .40 meter, excuse me, and v is 12 meter per second.

01:55 So now we have a period of 0 .032nd.

02:02 Okay, so, so that is a period...