Write the Equation Write the equation for a sinusoidal wave traveling in the negative direction

Write the Equation Write the equation for a sinusoidal wave...

Key Concepts

Wave Speed and Wavelength Relationship

The relationship between wave speed, frequency, and wavelength is given by v = f?. This fundamental connection allows one to determine the wavelength if the frequency and propagation speed are known, and it forms a cornerstone of understanding wave motion.

Direction of Wave Propagation

The sign in the wave equation (between kx and ?t) specifies the direction in which the wave propagates. For a wave traveling in the negative direction along the x-axis, the phase of the wave is expressed as kx + ?t, indicating that as time increases, the phase moves in the negative x direction. This concept is crucial for distinguishing between waves traveling in opposite directions.

Wave Number

The wave number, denoted as k, characterizes the spatial variation of the wave. It is defined as k = 2?/?, where ? is the wavelength. The wave number determines how many cycles of the wave fit into a unit distance, thus linking the wavelength to the spatial oscillation pattern.

Amplitude

Amplitude is a measure of the maximum displacement of the wave from its equilibrium position. In a sinusoidal wave, the amplitude represents the peak value of the oscillation and is a key indicator of the energy carried by the wave.

Sinusoidal Wave Equation

A sinusoidal wave equation describes a wave whose displacement varies sinusoidally with respect to both position and time. It is typically written in the form y(x, t) = A sin(kx ± ?t + ?), where A is the amplitude, k is the wave number, ? is the angular frequency, and ? is any phase shift. This form provides a systematic way to understand the oscillatory behavior of the wave in space and time.

Angular Frequency

Angular frequency, denoted as ?, quantifies how rapidly the wave oscillates in time. It is related to the frequency by the relation ? = 2?f, and it is essential for determining the time-dependent behavior of the sinusoidal wave.

Transcript

00:01 So this question we're asked to write the displacement equation for a wave, for a sinusoidal traveling wave, given that it's traveling along the negative x direction, and it has amplitude of 0 .01 meters, frequency 550 hertz, and a speed of 330 meters per second.

00:21 So that's write down everything that we're given.

00:23 So we're told that it's in the negative x direction.

00:30 So negative x direction we're told that it has a velocity of 330 meters per second that it has frequency of 550 hertz and it has an amplitude of 0 .01 meters so using all this we can determine the equation that describes the displacement of the wave.

01:20 So the base of this question is the equation that given in the book for the just for the wide displacement of the wave which is equal to the amplitude sign of kx plus or minus omega t and then we're going to assume there's no phase difference or phase term in the in the argument of the sign function.

01:47 So first of all since it's in the negative direction in the book negative negative extraction so i'm plot we're told that negative extraction implies that it's going to be kx plus omega t since plus always describes something and moving in the in the left direction and minus describes it moving in the in the right direction so that so that so now we have the sign of our of our of our function and we're also we know the amplitude that's just given to us so now we just need to determine what k in omega -half.

02:22 So we know that k is the wave number is equal to two pi over lambda and we don't have lambda but we can find it so we know that c is equal to f lambda which implies that lambda is equal to c over f which is equal to 330 meters per second over 550 hertz and if we work this out this is equal to 0 .6 meters so that means the wave number is equal to 2 2 pi over 0 .6 meters, which is equal to, if we work it out, it's equal to 10 .5 per meter.

03:07 So now that we have the wave number, all we need to find is the angular frequency...

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