(II) A sinusoidal wave traveling on a string in the negative...

(II) A sinusoidal wave traveling on a string in the negative $x$ direction has amplitude $1.00 \mathrm{cm},$ wavelength $3.00 \mathrm{cm},$ and frequency 245 $\mathrm{Hz}$ . At $t=0,$ the particle of string at $x=0$ is displaced a distance $D=0.80 \mathrm{cm}$ above the origin and is moving upward. (a) Sketch the shape of the wave at $t=0$ and (b) determine the function of $x$ and $t$ that describes the wave.

Key Concepts

Sketching Wave Shapes

Sketching a wave at a given moment involves plotting the displacement of the medium (such as a string) at various positions along the x-axis. This requires understanding how the sinusoidal function's parameters—amplitude, wavelength, and phase shift—translate into peaks, troughs, and zero crossings along the spatial domain. Accurate sketches help visualize wave behavior and are useful in comparing theoretical predictions with experimental observations.

Direction of Propagation

The sign before the term ?t in the wave function (either negative or positive) indicates the direction of the wave's propagation along the x-axis. A negative sign in the term (kx - ?t) typically denotes that the wave is moving in the positive x direction, whereas a positive sign would indicate motion in the negative x direction. Correctly identifying this sign is crucial for analyzing the travel direction of the wave.

Phase Constant and Initial Conditions

The phase constant ? is an adjustment in the wave equation that accounts for the initial conditions of the wave, such as its displacement and velocity at a given point at time t = 0. By choosing an appropriate phase constant, one can ensure that the wave equation accurately describes the observed motion of the wave, matching both position and direction of particle movement at specified instances.

Wave Number and Angular Frequency

The wave number, denoted by k, represents the number of radians per unit length and is given by k = 2?/?, where ? is the wavelength. Angular frequency, ?, represents how fast the wave oscillates in time and is given by ? = 2?f, where f is the frequency. These quantities allow the translation between spatial and temporal descriptions of the wave.

Amplitude, Wavelength, and Frequency

Amplitude is the maximum displacement of the wave from its equilibrium position and determines the energy or intensity of the wave. Wavelength is the spatial period of the wave—the distance over which the wave's shape repeats. Frequency is the number of oscillations per unit time. Together, these parameters define the scale and timing of the wave's oscillation and are interconnected through the wave speed.

Wave Equation

The wave equation for a harmonic sinusoidal wave typically takes the form y(x, t) = A sin(kx - ?t + ?) or y(x, t) = A cos(kx - ?t + ?). This equation expresses the displacement y as a function of both position x and time t, where A represents the amplitude, k is the wave number, ? is the angular frequency, and ? is the phase constant. This general form is fundamental for describing the behavior of traveling waves.

Sinusoidal Waves

Sinusoidal waves are periodic oscillations that can be mathematically described by sine or cosine functions. They are essential in physics for modeling a variety of wave phenomena, from vibrations on a string to electromagnetic waves. These waves are characterized by parameters such as amplitude, wavelength, frequency, and a phase constant, which helps define their spatial and temporal oscillatory behavior.

Transcript

00:01 In this question, we have a xoidal wave that is traveling on the string in the negative x direction, and it has an amplitude of 1 centimeter.

00:13 It has a couple of parameters, so let's just try to write down the expression for it.

00:26 It has an amplitude of 1 .00 cm.

00:31 So let's say everything is in terms of centimeter.

00:35 That is, the amplitude is 1.

00:38 It's sinusoidal.

00:42 Then we have kx minus omega -t.

00:46 Except that it travels in negative x -direction.

00:48 So this is actually plus omega -t.

00:51 So weight plan is 3 centimeter.

00:55 So lambda equals 3 centimeter.

01:00 Amplitude equals 1 centimeter.

01:04 Dumbda equals 3 centimeters.

01:06 So k equals 2 pi.

01:08 Over lambda and this gives us what k is.

01:14 So last one it says the frequency f equals 245 hertz and we know omega equals to pi f and that gives us what omega is.

01:28 So ht equals zero, so particle string and x equals zero is displaced by a distance 0 .8 centimeters.

01:42 So what that is saying is that d, t equals 0, x equals 0 .8 centimeter.

01:55 So it's moving upward.

01:59 What that says, that is to say this actually there is a face difference in the sign.

02:10 Or another way to think about this is that it has, it may, also have an offset.

02:20 So it's possible that you also have something like this plus a constant...

Please give Ace some feedback