At a given instant of time, two pressure waves, each moving at the speed of sound, emitted by a

step 1 Once again, welcome to a new problem. This time we're dealing with velocities and if you know ab

At a given instant of time, two pressure waves, each moving...

Key Concepts

Mach Number

The Mach number is a dimensionless quantity defined as the ratio of the speed of an object (or disturbance) relative to the speed of sound in the surrounding medium. It plays a crucial role in compressible flow dynamics by distinguishing between subsonic, transonic, and supersonic regimes, and it is used to determine the behavior of pressure waves and the formation of shock waves or Mach cones around moving objects.

Mach Cone

A Mach cone is the conical envelope formed by the coalescence of pressure waves generated by an object moving at supersonic speeds. Its half-angle is determined by the Mach number, with higher speeds leading to narrower cones. This concept is key in studying shock waves, as the angle of the cone provides direct insight into the relationship between the object's speed and the speed of sound in the medium.

Pressure Wave Propagation

Pressure waves, generated by disturbances such as a moving point source, propagate through a medium at a characteristic speed known as the speed of sound. Understanding the propagation of these waves is essential in fluid dynamics, as it allows for the analysis of phenomena like the interference of successive wavefronts, the build-up of wavefronts into a shock wave, and the spatial relationship between the source and the observed pressure disturbances.

Transcript

00:01 Once again, welcome to a new problem.

00:03 This time we're dealing with velocities and if you know about the match number, the match number has a ratio where at the top you're thinking of the velocity of a specific fluid relative to the velocity of sound.

00:27 So this is the match number and a lot of times there is always going to be an interface between sound and the velocity of the fluid and in terms of match numbers you could talk about march 1 where march 1 reflects the speed of sound at sea level speed of sound at sea level and that turns out to be 760 miles per hour and of course march 2 i don't i don't know why i'm saying march hasn't the the month no not the month just march numbers so march 2 would be twice this speed twice the speed of sound in in fluids like this, in fluids like this.

01:35 So coming back to our problem, we're given a specific time.

01:43 So we're going to say at an instance, we have two pressure waves.

01:55 This is at an instance, we have two pressure waves.

01:58 And each one is moving at the speed of sound, each one is moving at the speed of sound.

02:10 So there's a specific point right there.

02:13 These are the centers of the waves.

02:15 This is the center of the first wave and the second wave.

02:20 At a radius, the fast wave is at a radius of two inches.

02:25 And the second wave is at a radius of 10 inches.

02:29 And the gap between the centers happen to be five inches.

02:37 That's the gap between the centers, happen to be five inches.

02:41 And these are two pressure waves.

02:45 And they result from a point source.

02:52 They result from a point source.

02:54 You know, we can assume the point source is inside of this.

02:57 So this is the point source that's producing the sound that's producing the sound itself.

03:07 And were given, given that the point source, given that the point source moves with constant speed.

03:30 So constant speed means that you don't really have any acceleration.

03:35 And the fluid is at rest.

03:39 So, you know, the speed of sound, we talked about the march number, the velocity of the fluid is v fluid is zero because it's at rest.

03:52 And then, of course, the sound, v sound is constant.

03:57 So speed moves with constant velocity.

04:00 So goal, and this problem is twofold, one, to determine the match.

04:13 Number and then the second one visualize the location of the point source so visualize the location of the point source those are the two things we're looking at in this particular problem we're looking at the fact that we want to get the match number and then we also want to visualize the location of the point source in terms of equations in terms of equations the march number comes out as the velocity of the wave.

05:08 And this is the velocity, you know, coming back, you see we had the velocity of the fluid, the velocity of the fluid in the field, divided by the velocity of sound in the field.

05:32 So this is fluid velocity and this is sound velocity, sound velocity in the field.

05:52 Fluid velocity and sound velocity in the field.

05:56 Those are the two things we're looking at.

05:59 So in terms of diagrams, we're saying that we have the bigger velocity right there, and then we also have the smaller waves, so the bigger wave.

06:15 And the smaller wave, this is the point source p.

06:19 So p is the point source.

06:24 And then also we do have the centers of these two waves.

06:29 The first one, as you said before, happens to be at two inches.

06:34 The second one happens to be at 10 inches.

06:39 And then the gap between the two, this is.

06:43 02, this is 01, the gap, this gap between the two is the same as velocity times the time of the wave as the fluid and then that distance is 5 inches.

07:07 That distance is 5 inches.

07:10 Recall that the reason why we're using ct and vt, the distance, the distance is speed times time.

07:23 So in the first case you're saying the speed is v and then the time is t of wave and then in the second case you're saying c and then the time of the wave.

07:36 So those are the distances travelled by the two waves that we're dealing with.

07:41 And so from the figure you get to see that the bigger wave, which is a sound wave, is going to be ct is 10 inches.

07:49 This is the sound wave.

07:54 And so c, t minus t of wave is two inches.

08:06 It's two inches.

08:07 So we're getting the differences between the two.

08:11 The first case is the velocity of the bigger wave...

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