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Lecture 29

Goals:

- Chapter 20, Waves

- Final test review on Wednesday.
- Final exam on Monday, Dec 20, at 5:00 pm.
- HW 11 due Wednesday.

Exercise Wave Motion

- A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 meters and the speed of the waves is 5 m/s, how long Dt does it take the boat to go from the top of a crest to the bottom of a trough ? (Recall T = / v )

(A) 2 sec(B) 4 sec(C) 8 sec

t

t + Dt

Mathematical formalism

D(x=0,t)

D(0,t) ~ A cos (wt + f)

- w: angular frequency
- w=2p/T

t

T

λ

D(x,t=0)

D(x,0) ~ A cos (kx+ f)

- k: wave number
- k=2p / l

x

Mathematical formalism

- The two dimensional displacement function for a sinusoidal wave traveling along +x direction:

D(x,t) = A cos (kx - wt + f)

A : Amplitude

k : wave number

w : angular frequency

f : phase constant

Mathematical formalism

- Note that there are equivalent ways of describing a wave propagating in +x direction:

D(x,t) = A cos (kx - wt + f)

D(x,t) = A sin (kx - wt + f+p/2)

D(x,t) = A cos [k(x – vt) + f]

Why the minus sign?

- As time progresses, we need the disturbance to move towards +x:

at t=0, D(x,t=0) = A cos [k(x-0) + f]

at t=t0, D(x,t=t0) = A cos [k(x-vt0) + f]

vt0

v

x

Which of the following equations describe a wave propagating towards -x:

D(x,t) = A cos (kx – wt )

D(x,t) = A sin (kx – wt )

C)D(x,t) = A cos (-kx + wt )

D) D(x,t) = A cos (kx + wt )

Speed of waves

- The speed of mechanical waves depend on the elastic and inertial properties of the medium.

- For a string, the speed of the wave can be shown to be:

Tstring: tension in the string

m=M / L : mass per unit length

Waves on a string

- Making the tension bigger increases the speed.
- Making the string heavier decreases the speed.
- The speed depends only on the nature of the medium, not on amplitude, frequency etc of the wave.

Exercise Wave Motion

- A heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom.
- As the wave travels up the rope, its speed will:

v

(a) increase

(b) decrease

(c) stay the same

Sound, A special kind of longitudinal wave

λ

Individual molecules undergo harmonic motion with displacement in same direction as wave motion.

Plane waves

- Note that a small portion of a spherical wave front is well represented as a “plane wave”.

Intensity (power per unit area)

- A wave can be made more “intense” by focusing to a smaller area.

I=P/A : J/(s m2)

R

Exercise Spherical Waves

- You are standing 10 m away from a very loud, small speaker. The noise hurts your ears. In order to reduce the intensity to 1/4 its original value, how far away do you need to stand?

(A) 14 m (B) 20 m (C) 30 m(D)40 m

Intensity of sounds

- The range of intensities detectible by the human ear is very large
- It is convenient to use a logarithmic scale to determine the intensity level,b

I0: threshold of human hearing

I0=10-12 W/m2

Intensity of sounds

- Some examples (1 pascal 10-5 atm) :

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