A loudspeaker with a diaphragm that vibrates at 960 Hz is traveling at 80.0 m/s directly toward

step 1 Okay, let's begin solving the problem 48. So it said this, if we have one loudspeaker and it's r

A loudspeaker with a diaphragm that vibrates at 960 Hz is...

Key Concepts

Doppler Effect

The Doppler Effect describes how the observed frequency (and therefore wavelength) of a wave changes when the source and observer are in relative motion. When the source is moving toward the observer, the waves are compressed, leading to a higher observed frequency, while moving away results in a lower frequency. This concept is essential when dealing with moving loudspeakers or other emitting sources in sound and light phenomena, as it influences the interference conditions by altering the effective wavelength of the waves in the observer’s frame.

Interference of Waves

Interference of waves occurs when two or more waves superpose to form a resultant wave of greater, lower, or the same amplitude. In the context of sound waves passing through multiple openings or slits, interference causes regions of constructive interference (amplification) and destructive interference (cancellation). The condition for destructive interference is typically determined by the path difference between the waves, which must equal an odd multiple of half wavelengths for cancellation to occur.

Wavelength and Frequency Relationship

The relationship between wavelength, frequency, and the speed of propagation of a wave is given by the equation: speed = frequency multiplied by wavelength. This fundamental relationship allows one to calculate the wavelength from a known frequency and propagation speed, which is critical when analyzing interference patterns in acoustics and other wave phenomena.

Geometric Path Difference in Interference

The geometric path difference is a measure of the extra distance one wave travels relative to another when reaching a particular point in space. In interference experiments, such as those involving sound passing through two openings, the path difference is related to the separation between the openings and the observation angle. The condition for the first destructive interference (first cancellation) is commonly expressed as the path difference equaling one half of the wavelength, linking the geometry of the setup to the interference pattern observed.

Transcript

00:02 Okay, let's begin solving the problem 48.

00:04 So it said this, if we have one loudspeaker and it's running towards the wall here with a velocity 80 meters per second, and by the meantime, it makes the voice and the frequency of the voice is 960 hertz.

00:32 And at this temperature in the air, the normal speed of the voice is 344 meters per second.

00:43 And on the wall, there are two holes there.

00:46 There's a hole one, there's a hole two, and the distance between the two holes, we call a here.

00:54 You are supposed to standing at some position here.

00:59 And at this line, which is the point, parallel to the wall you moved up and down and found two special places as those two places the voice cancels which means you hear nothing there and you find that the angular position of those two special point equals plus 11 .4 degree or minus 11 .4 degree so um so um our question is, what will be the width between the whole one and the whole two here? we call that a here.

01:44 And the question two is, if the velocity of the speaker equals zero, what will be the new theta, which means at those angular position, the new angular position, you hear nothing or say the sound just canceled with each other.

02:03 So to solve this problem, first of all, we need to basically introduce the doppler effect, calculate the effective wavelength of the speed.

02:17 So what will be the effective lambda here? if we have a source which didn't move, it will be simply the velocity divided by the frequency.

02:37 It equals this.

02:38 For us, it equals 344 divided by 960.

02:44 We make sure all the units is in the international system and it will roughly equal to 0 .3, 7, oh not 7, 0 .3, 58 emitters.

03:07 But here we notice that we have a voice source is traveling, it's traveling, and the velocity of the source equals 80 meters per second.

03:22 So here we have new wavelength, we call the lambda prime, it will equal the lambda minus, the velocity of source towards the audience or say maybe you.

03:40 You are standing on the ground divided by its original frequency.

03:51 So it equals 0 .275 meters.

04:00 So we see that this is called doppler effect.

04:10 It dramatically changed the wavelength of the voice from the speaker.

04:17 So we will just use this lambda prime to calculate the first case, which is we have the theta equals 11 .4 degree.

04:35 So then we go back to use the equation of interference.

04:44 So i printed here, the sine theta actually equals plus minus.

04:52 We've lost lamp over divided by a here.

04:56 So here because we consider it as double effect, so lamp should be a lambda prime here...