Transverse waves on a string have wave speed 8.00 m / s, amplitude 0.0700 m, and wavelength 0.

Transverse waves on a string have wave speed 8.00 m / s,...

Transverse waves on a string have wave speed $8.00 \mathrm{~m} / \mathrm{s}$, amplitude $0.0700 \mathrm{~m},$ and wavelength $0.320 \mathrm{~m}$. These waves travel in the $x$ direction, and at $t=0$ the $x=0$ end of the string is at $y=0$ and moving downward. (a) Find the frequency, period, and wave number $k=2 \pi / \lambda$ of these waves. (b) Write the equation for $y(x, t)$ describing these waves. (c) Find the transverse displacement of a point on the string at $x=0.360 \mathrm{~m}$ at time $t=0.150 \mathrm{~s}$

Key Concepts

Wave Equation for a Traveling Wave

The wave equation describes the transverse displacement of a point on the medium as a function of position and time. A common form is y(x, t) = A sin(kx - ?t + ?) or y(x, t) = A cos(kx - ?t + ?), which encapsulates the wave’s amplitude, spatial variation, time evolution, and phase shift.

Phase Constant Determination

The phase constant (or phase shift) is a parameter in the wave equation that adjusts the trigonometric function to match the initial conditions of the physical situation, such as the initial displacement or velocity of the wave at a given point.

Angular Frequency

Angular frequency quantifies how rapidly the wave oscillates in time and is defined by ? = 2?f. It is used in the analysis of wave motion, particularly in sinusoidal functions that represent the wave’s displacement.

Wave Number

The wave number is a measure of the number of radians of phase per unit distance and is defined as k = 2?/?. It provides a way to describe how rapidly the wave oscillates in space.

Frequency and Period

Frequency is the number of complete oscillations or cycles per unit time, typically measured in hertz (Hz), while the period is the time taken for one complete cycle. They are inversely related, such that T = 1/f, where T is the period.

Wave Speed

Wave speed is the rate at which a disturbance travels through a medium. It is defined as the distance traveled per unit time and relates directly to the wavelength and frequency, following the relationship v = ?f.

Transcript

00:01 So in this question we have wave speed b equals to 8 meter per second and amplitude a equals to 0 .070 meter and wavelength lambda equals to 0 .302 320 meter.

00:15 Okay so moving to the part a in which we have to determine the wave frequency.

00:20 So we know that v equals to lambda multiplied by f.

00:26 So from here, substituting the values, we get 8 meters equals to 0 .320 multiplied by f.

00:35 So frequency f comes out to be 25 .0 hertz.

00:40 Okay? so this is the answer for the frequency.

00:43 Now, in the same part, we have to determine the time period, which is equals to 1 by frequency.

00:49 So 1 by 25 seconds.

00:52 So from here, time period comes out to be 0 .040 second.

00:56 Okay, this is the answer for the time period.

00:59 Now in the same part we have to determine the wave number k and which is equals to 2 pi by lambda.

01:07 So substituting the value of lambda we get 2 pi by 0 .320.

01:12 So from here we get k equals to 19 .6 meter inverse.

01:17 So this is the answer for the wave number of the part a.

01:21 Now moving to the part b in which we have to determine the wave function.

01:25 So wave function will be given by y x comma t equals to a cos 2 pi x by lambda minus plus t by capital t okay so now substituting the values we get we get y x comma t equals to 0 .070 cos 2 pi x by x by 0 .320 minus plus t divided by 0 .040...

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