A sound wave is described by D(y, t)=(0.0200 mm) ×sin[(8.96...

Key Concepts

Graphical Representation of Wave Displacement

Plotting displacement versus time for a fixed spatial position involves substituting a constant value for the spatial variable into the wave equation. This graph illustrates how the displacement of the medium (e.g., air molecules in a sound wave) varies with time at that particular point, highlighting the periodic nature of the oscillations and the amplitude of the wave.

Wavelength and Wave Speed

The wavelength is related to the wave number by the relation ? = 2?/k and represents the distance over which the wave's shape repeats. The wave speed is given by v = ?/k and indicates the rate at which the wave phase propagates through the medium. These parameters are fundamental to understanding the spatial and temporal evolution of the wave.

Period of Oscillation

The period of the wave, defined as T = 2?/?, represents the time required for one complete cycle of the wave oscillation. The period, together with the wave speed and wavelength, ties together the temporal and spatial characteristics of the wave, providing a comprehensive understanding of its dynamics.

Wave Propagation Direction

The direction in which the wave travels is determined by the signs of the spatial and time-dependent terms in the phase. In the standard sinusoidal form, a phase given by (ky ? ?t) denotes propagation in the positive direction of the spatial coordinate, while a phase of (ky + ?t) indicates propagation in the negative direction. Recognizing this helps in understanding how waves advance through a medium.

Wave Equation Representation

The wave is expressed as a sinusoidal function of both a spatial variable and time. This representation typically takes the form A sin(ky ± ?t + ?), where A is the amplitude, k is the wave number, ? is the angular frequency, and ? is the phase constant. This form encapsulates all the essential properties of the wave including its magnitude, phase, and variation in space and time.

Particle Oscillation Direction

For waves, it is important to distinguish between the direction of the wave’s propagation and the direction in which the medium's particles oscillate. In sound waves, which are longitudinal, the oscillation of the air molecules occurs in the same direction as the wave is traveling. This relationship contrasts with transverse waves, where the medium’s displacement is perpendicular to the direction of propagation.

Transcript

00:01 So once again i will come to a new problem.

00:06 We're still looking at sound waves and a lot of these waves are simeithoidal so these are transversal waves and it so happens that we have a a y -axis and then we have the time axis, the t -axis, the time axis.

00:29 So if the wave itself is given in terms of y and t, it's going to be y and t being the parameters.

00:44 So we end up having 0 .00 millimeters times sign off 8.

00:58 Well, i guess we're going to have to fit this on the screen because it's a long equation.

01:07 Each wave has its own equation, so this one we want to fit it right here.

01:12 It's given by 0 .200 millimeters times sine of 8 .96 radiance.

01:28 Remember radiance is the measurement for angles, the different kind of measurement for angles.

01:35 This is y.

01:36 Plus, again, this is a long equation.

01:39 So you want to bear with me 3140 radiance per second, and that's the time factor.

01:47 So we have the displacement factor in the wide direction and the time factor.

01:52 And then we have the face angle, which is pi over four radiance.

01:58 Pi over four radiance.

01:59 Remember the time itself, this time itself, is measured in seconds.

02:07 That's the time it's measured in seconds.

02:09 There are three things we have we have a we want to find out the wave direction that's something we want to find out like which direction given this equation can you tell us how the wave is moving and then the second thing is we want to find the axis of oscillation meaning how is the wave bouncing up and down of back and forth.

02:43 That's what we see with the oscillations.

02:46 We're also looking for lambda, which is the wavelength.

02:50 That's one of the things we're looking for.

02:53 We're looking for the speed of the wave.

02:59 We're looking for the speed of the wave.

03:02 And then we're also looking for omega.

03:05 This speed of the wave, you can call it the v of velocity.

03:09 Omega is the period.

03:16 Actually omega would be a different thing, that would be the frequency.

03:21 So in this sense we want to say we're looking for the period, which is capital t.

03:27 This is the period.

03:29 This is capital t.

03:32 That's part c and part d.

03:36 We want to find displacements and not just displacements.

03:45 We want to relate displacements to the time from a graphical point of view and d equals to at y these are these are numbers that we're dealing with y one meter and then at y equals to one meter we're looking at the times when it's zero seconds up until the times when it's four milliseconds so that's that's what we're dealing with in this particular problem.

04:24 So there are four sections that we want to go through in the problem and this elicits different kinds of understanding.

04:32 So we have different kinds of understanding when it comes to this problem.

04:37 The first thing you're saying is think about a wave function.

04:43 Think about a wave function that's sinusoidal and it's moving in the x direction.

04:53 The positive x direction means that it's moving towards the right.

04:57 If you want to display the template, you want to call it a template for the wave equation, xt, a -tine 2 -pi, remember a is the amplitude that we're dealing with.

05:18 This is a template that you want to come across when you're solving problems involving waves.

05:25 So x over lambda minus t, which is time and capital t, that's the period.

05:33 And then we also have phi not.

05:36 And this simplifies to a sine kx minus omega -t plus phi -0.

05:47 And so there are certain values that are critical to understanding this problem, including k, which is a wave constant, is 2 pi over lambda, and then omega is 2 pi over t.

06:05 Remember, omega is the angular frequency.

06:09 That's the angular frequency or angular velocity.

06:15 I want to call it that.

06:19 So that's what you're seeing right there.

06:22 There are certain highlights we have to make about this problem.

06:25 And why is the displacement distance in the perpendicular in the perpendicular direction.

06:45 So meaning bouncing, bouncing up and down in the wide direction.

06:50 That's what you're saying even though the waves is propagating.

06:54 Then of course a is the amplitude of the waves.

06:58 You have to understand the issues involving this wave x is the equilibrium position of the wave so if you have a y axis right there and you have an x axis and you have waves that are moving this position right here this is this is your x axis and that's what we call you your equilibrium position so the displacements up and down balance out kind of like balance out t is the time period, t is the time period, and lambda is the wavelength.

07:44 Remember, phi is the face angle, or the face shift, or the face angle.

07:52 That's how a phi.

07:54 The position, if you have a graph and the graph as an x -axis and the y -axis, and you have a sine of a third wave, going through the graph there are certain issues that you obviously are going to detect in this particular problem obviously the equilibrium position is a straight line and that's the x -axis as you can see when when the wave is moving and think about any particle on the wave that particle is going to be displaced relative to the equilibrium position and as the wave is moving it could be off the equilibrium position like in this case or it could be on the equilibrium position like in the second case so this would be like a and this is or maybe i can use other variables since we've already used a so i can say maybe instead of x y z i can take m so m and then this is n and then this is n and then this is o.

09:11 So these are different positions of a particle.

09:15 And knowing that the equilibrium position is x, the displacement is going to happen relative to the y.

09:25 And the value of y will be dependent on the position.

09:30 So for example, the particle m has a positive y value.

09:36 The particle n has a neutral value.

09:41 And then the particle o has a negative y fad.

09:46 So those are values of y relative to the x.

09:52 And the value of y depends on that.

09:56 And it also depends on the time periods.

09:59 So this would be a time at m and this is a time at n and this is a time at o.

10:05 So these are different time periods.

10:08 And you can see that obvious rule, y is going to be a function of x and t.

10:15 Y is going to be a function of x and t.

10:18 X being position relative to the equilibrium, and t being the time period.

10:31 So that's what you're looking at.

10:33 Why is the function? so if you know this function and you're looking at the wave motion, you could always talk about the displacement from the equilibrium.

10:46 So first of all, you're talking about the position.

10:50 You're talking about the position of a particle relative to the equilibrium, which is this, and that in turn can help you determine the displacement.

10:59 Can help you determine the displacement.

11:01 So these are issues you're dealing with.

11:04 So we can use the general form of the web equation to solve this, problem and there are different ways you could do that the different ways you could do that so in this sense since we know that y is dependent on the position x in the time period this is going to give us an equation of the wave moving in the x direction relative to time and starting a specific face angles so we can simplify that to a sine kx minus omega t so it's finite and then the second this would be this would be kind of like our first equation the position equation the second equation you're looking at is the displacement equation that we've been given in the problem and we already had it before we're just writing the same equation radium per meter and this is y and plus 314 radian for second time is pi over four radiance so in going back to part a remember they were asking you know what's the wave direction what's the wave direction the wave oscillates in the direction that um if you think about the wave itself, the wave is traveling, or we want to say this.

13:09 Given this equation, this equation is in terms of why, so it means that the wave, in part a will say the wave is traveling.

13:22 It's traveling in the, or not in the, but along the y axis.

13:36 And the hint for that is the fact that they're giving us this why.

13:40 The wave is traveling along the y -axis.

13:48 The wave is traveling along the y -axis...

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