FIGURE EX20.42 is a snapshot graph at t=0 s of a 5.0 Hz wave...

Key Concepts

Wave Speed

Wave speed is a fundamental concept in wave mechanics, defined as the distance a wave travels per unit time. It is determined by the product of the wave’s frequency and its wavelength. In practice, if the wavelength is known from a spatial graph (like a snapshot of the wave), multiplying it by the frequency yields the wave speed.

Frequency and Wavelength

Frequency is the number of oscillations or cycles per second, while the wavelength is the distance over which the wave's shape repeats. The relationship between frequency and wavelength is key because it allows for the calculation of wave speed, and it is also used to derive other important parameters such as angular frequency and the wavenumber.

Angular Frequency and Wavenumber

Angular frequency (?) is obtained by multiplying the frequency by 2? (? = 2?f), and represents the rate of change of the wave's phase in radians per second. The wavenumber (k) is calculated as 2? divided by the wavelength (k = 2?/?), indicating how many radians the phase changes per unit distance. Both of these quantities are essential in formulating the mathematical expression of the wave.

Phase Constant

The phase constant is a term added to the argument of the sinusoidal function describing a wave to account for any initial phase offset at a specific position and time, typically at the origin. It ensures that the mathematical model accurately reflects the physical configuration of the wave at an initial condition.

Displacement Equation of a Traveling Wave

A traveling wave is generally represented by an equation of the form y(x,t) = A sin(kx ± ?t + ?), where A is the amplitude, k is the wavenumber, ? is the angular frequency, and ? is the phase constant. The choice of plus or minus in the argument of the sine function indicates the direction of wave propagation; for example, a minus sign corresponds to a wave traveling in the positive x direction, while a plus sign indicates travel in the negative x direction.

Transcript

00:01 So, you know, once again, we're back to the same type of problems that we have been dealing with.

00:11 And this means that we have to sketch a graph.

00:16 The x -axis is obviously in meters, and the displacement is in millimeters.

00:23 That's the displacement of a wave.

00:26 So this is a wave that's traveling towards the left.

00:34 So this wave is traveling towards the left.

00:36 And, you know, we have to try and sketch it to try and sketch it to relevant standards.

00:43 Try and sketch it to relevant standards.

00:46 And so if you look at the wave, you see that it's going to have a velocity towards the left and then it's divided into meters.

01:03 So this is two, and these are approximate values.

01:07 So it's not the exact values.

01:10 If we had to have software tools, then we can develop exact values, but for purposes of solving this particular problem, we sketch it to its estimates.

01:26 So right here we have 0 .5 and that's in millimeters and then this is negative one.

01:32 This is also negative because it's below the x -axis.

01:35 At the top we have 1.

01:38 Well actually we have 0 .5 not 1.

01:42 This is 0 .5 right here.

01:46 And then we're still progressing and we have 1 right there.

01:52 All this is in millimeters for the graph and it's moving towards.

01:58 The left there are three requirements the first one is we have to determine the velocity of this graph as it moves towards the left and then the second thing is we have to determine the the face constant the face constant which we call k if you can look at this graph you see that it doesn't start at the origin so there's some type of face displacement and so we have to account for that.

02:31 We have to accommodate for that.

02:34 And then finally, we need to write the displacement equation, which is a product of two things.

02:47 It's a product of the movement in the x direction, and it's also a product of the time in seconds.

02:56 I remember the movement in the x direction that's measured in meters.

03:04 That movement is going to be measured in meters.

03:09 And so the movement or the position, you want to call the x the position, that's relevant to solving the problem.

03:18 The given information includes the fact that the graph starts at t equals to zero.

03:26 And the frequency is measured in hertz at 5 .0.

03:32 And so in getting the speed in part a, we get to see that the velocity is f lambda, where lambda is the wavelength, and we can always get that lambda from the graph itself.

03:52 So we're saying if you look at the graph, what's the position? you know, between these two crests or between these two troughs.

04:10 If you look at that, it's approximately two meters, so we're going to go with that as our lambda.

04:16 So lambda equals to two meters and then remember the frequency was 5 .0 hearts.

04:22 And so our velocity is your frequency which is 5 harts times the lambda, which is our wind, that's 2 meters.

04:36 This comes out to a velocity of 5 times 2 which is 10 meters per second.

04:43 That's the first part of the problem.

04:45 The second part is where we had to determine the displacement, we had to determine the displacement, or rather not the displacement, but the face constant.

04:58 So our goal right here is to determine the the face constant.

05:13 Okay, you have to determine the face constant.

05:20 So, you know, first of all, we have to get the face.

05:25 We have to get the face.

05:28 And so we get to see that a couple of things are going to happen here.

05:37 So remember, k, remember we have dxt, that's our wave.

05:45 A sine theta or rather sign kx minus omega t plus finite a is the amplitude k is the wave number x is the position omega is the angular velocity omega is the angular velocity or rather angular frequency we'll call it that angular frequency that's omega.

06:33 And then finite is the face difference.

06:42 Finot is the face difference.

06:44 That's what we're contending with in this particular problem.

06:49 If you look at the graph, if you go back to the graph, you see this position is zero.

06:54 So that's going to be helpful because now when t equals to zero, the position is also at zero.

07:04 And the displacement of the wave is 0 .5 millimeters.

07:10 And you can always confirm that at this point, this is the displacement of the graph right here.

07:21 And that happens when the time is zero and also when the position is zero.

07:30 That's the displacement we're dealing with.

07:34 So, you know, the question is why is that helpful in solving the problem? well, of course, it's very helpful because we're going to plug in numbers relevant to helping us solving the problems using this equation.

07:48 So using the equation, we know that d is 0 .5 millimeters and our amplitude is 1 millimeter.

07:58 If you go back, you see that this is one millimeter.

08:04 That gap right there, that amplitude, this is our a.

08:08 That's one millimeter.

08:14 And then, of course, we have sign of k.

08:19 K is going to be multiplied by zero because x is zero.

08:25 And then omega, the time is also zero, right here...