Wave Speed and the Doppler Effect

15 kg

0.5 m

Whiteboard Warmup!

A 15-kg block is hung from a 0.5-m long string of mass 3 g. When the string is plucked, it produces a wave that is shown at one instant to the left.

What will be the frequency of the resulting note?

15 kg

0.5 m

λ = 1m

FT = (15 kg)(10 m/s2) = 150 N

μ =M/L = ( 0.003 kg)/(2 m)

μ = 0.0015 kg/m

v = 316.2 m/s

v = λf

f = 316.2 Hz

Whiteboard Reasoning

A sound wave produced by a 300 Hz source travels at ≈343 m/s in air at room temperature.

How fast would a sound wave produced by a 150 Hz source travel in the same room?

The same amount!!

The speed of a wave depends only on the medium.

The Doppler Effect!

The Doppler Effect is the shift in the observed frequency of a wave, based on the relative

velocity of the source and the observer.

This shows a source that is moving to the right with a constant velocity.

Observer A has a relative velocity away from the source, and will hear a lower frequency than the emitted frequency.

Observer B has a relative velocity toward the source, and will hear a higher frequency than the emitted frequency.

If the source and the observer are moving closer relative to one another, the observer will perceive a higher frequency than is emitted.

If the source and the observer are moving further away relative to one another, the observer will perceive a lower frequency than is emitted.

The Doppler Effect only pertains to frequency. A common misconception is that the Doppler Effect changes the volume of the perceived sound. This is not so!

The intensity (volume) only depends on the distance from the source, and is not a part of the Doppler Effect

A small vibrating object on the surface of a ripple tank is the source of waves of frequency 20 Hz and speed 60 cm/s. If the source S is moving to the right, as shown above, with speed 20 cm/s, at which of the labeled points will the frequency measured by a stationary observer be greatest?

(A) A (B) B (C) C (D) D

(E) It will be the same at all four points.

Doppler Whiteboard!

In the Doppler effect for sound waves, factors that affect the frequency that the observer hears include which of the following?

I. The speed of the source II. The speed of the observer III. The loudness of the sound

 (A) I only (B) III only (C) I and II only

(D) II and III only (E) I, II, and III

Doppler Whiteboard!

A small vibrating object on the surface of a ripple tank is the source of waves of frequency 20 Hz and speed 60 cm/s. If the source S is moving to the right, as shown above, with speed 20 cm/s, what will be heard by observer A?

(A)A frequency that is lower than 20 Hz and decreasing.(B)A frequency that is lower than 20 Hz and constant.(C)A frequency that is lower than 20 Hz and increasing.(D)A frequency that is exactly 20 Hz and constant.(E)Observer A will not hear anything

Final Doppler Whiteboard!

Observer A will hear a constant frequency that is lower than the emitted frequency.

Observer C will hear a constant frequency that is higher than the emitted frequency.

Moving at the Speed of the Waves!

In this case, the source actually stays adjacent to each of its emitted waves, creating a large build-up wavefronts directly along its motion.

Supersonic Motion!

The source actually outruns each of its emitted waves, creating a large cone of built-up wavefronts known as a sonic boom.

In the case of sound waves, this would be a large high-pressure area where all of the wavefronts overlap.

Boundary Reflections and Intensity

A small vibrating object S moves across the surface of a ripple tank producing the wave fronts shown above. The wave fronts move with speed v. The object is traveling in what direction and with what speed relative to the speed of the wave fronts produced?

Direction Speed(A) To the right Equal to v(B) To the right Less than v (C) To the right Greater than v(D) To the left Less than v (E) To the left Greater than v

Boundary Reflections

When a wave reflects off of a more dense medium than the one in which it is traveling, it will become inverted.

When a wave reflects off of a less dense medium than the one in which it is traveling, it will not become inverted.

A square wave pulse is incident on a fixed boundary, as shown below.

Draw the shape of the string 2 seconds later.

Boundary Reflections and Superposition: Unite!!!

Red: Incident wave contribution

Blue: Reflected wave contribution

2 seconds later…

Net Wave:

A square wave pulse is incident on an open boundary, as shown below.

Draw the shape of the string 2 seconds later.

One more time!

Red: Incident wave contribution

Blue: Reflected wave contribution

2 seconds later…

Net Wave:

The string shown above is fixed at one end, and loose on the other. The pulse shown above is incident on the fixed end. How many reflections will it make before it returns to the state (position and velocity) shown above?

(A) One (B) Two (C) Three (D) Four (E) Five

Dense boundary: Inversion

Non-dense boundary: No Inversion

Dense boundary: Inversion

Non-dense boundary: No Inversion

Sound Intensity (I)

A measure of the loudness of a sound wave.

Units are Watts per square meter (W/m2)

The volume of a sound wave is governed by the amount of energy that passes through a given area per second.

Sound Intensity!When sound is emitted by a source, it travels outward in all directions.

This is called a spherical wave.

BUT, since the sound wave travels outward in all directions, but has a set amount of total power, this means that the power of the sound wave is spread out over a larger and larger sphere as it gets further away!

Asphere = 4πr2

You have the power emitted by the source, but spread out evenly over an ever-expanding sphere of sound.

I α 1/r2

Intensity has an inverse-squared dependence on distance from the source of the sound.

You can think of it as the sound wave “spreading out” over the surface of a sphere.

The beauty of the inverse-squared relationship is so great that some guy actually got it tattooed on his arm.

What you need to know

Sound intensity is proportional to 1/r2, where r is the distance to the source of the sound.

Any type of quantity that spreads outward in all directions (gravitational field, electric field) will

have an inverse-squared relationship.

You must know how to proportionally reason using the inverse-squared relationship.

Butter Gun WhiteboardAlbert Pujols connects with a fast ball and sends it out of the park. A fan in the outfield bleachers, 140 m away, hears the hit with an intensity of 2.2 10-6 W/m2. Assuming no reflected sounds, what is the intensity heard by the first-base umpire, 90 ft (27.4 m) away from home plate?

Solution

I α 1/r2

Since the umpire is (27.4 m)/(140 m) = 0.2 times the distance from the source of the sound, he will hear an intensity that is 1/(0.22) = 25 times as large!

2.2 10-6 W/m2 * 25 = 5.5 10-5 W/m2

Inverse-Squared Quiz!

At an Occupy Ridgedale Avenue protest, a loudspeaker is producing sound waves that spread out in all directions. Protestor A is 5 meters from the loudspeaker, and receives sound of intensity 2.5 × 10-3 W/m2. Protestor B is 10 meters from the loudspeaker. What sound intensity does she receive?

r is multiplied by 2

I is multiplied by 1/(22) = 1/4

Therefore, I = (1/4)*(2.5 × 10-3 W/m2) =

6.25 × 10-4 W/m2