Lecture 38Review of wavesInterference

Standing waves on a string

What is a wave ?

Examples:– Sound waves (air moves back & forth)– Stadium waves (people move up & down)– Water waves (water moves up & down)– Light waves (what moves??)

A wave is a traveling disturbance that transports energy but not matter.

Waves exist as excitations of a (more or less) elastic medium.

Amplitude: The maximum displacement A of a point on the wave.

Amplitude A

A

Period: The timeT for a point on the wave to undergo one complete oscillation.

x

y

A few parameters

1f

TFrequency: Number of oscillations f for a point

on the wave in one unit of time.

Angular frequency: radians ω for a point on the wave in one unit of time.

2 f

y

Wavelength: The distance λ between identical points on the wave.

Speed: The wave moves one wavelength λ in one period T, so its speed is

Connecting all these simple harmonic motions

Wave number k :

x

λ

Propagation

Moves in the +x direction with v

Moves in the −x direction with v

Examples: Harmonic waves

The wave equation

General wave: y f x vt

2 22

2 2

y fv

t uy f

vt u

Let u x vt

2 2

2 2

y fx uy f

x u

2 22

2 2

y yv

t x

Wave equation

2 2

2 2 2

10

y yx v t

Wave energy

• Work is clearly being done: F.dr > 0 as hand moves up and down.

• This energy must be moving away from your hand (to the right) since the kinetic energy (motion) of the end of the string grabbed by the hand stays the same.

P

Power

Energy for a particle in SHM (attached to a spring k)

2 2 212

E kA A 2 km

This energy propagates at speed v.

the average energy per unit time that flows in the direction of propagation should be proportional to v

2 2P v A Average power

Intensity

Example: A siren emits a sound of power 2W at 100 m from you. How much power reaches your ear (eardrum area = 0.7 cm2)Intensity at distance r from source:

5 2at source

2 2

2 W1.6 10 W/ m

4 4 100 mR

PI

r

area

PI

Average power (over time) in wave

Area of the surface where this power is distributed

5 2 4 2eardrum area of eardrum 1.6 10 W/ m 0.7 10 m 1.1 nWRP I

Power absorbed by eardrum:

r

2

Sphere of

area 4 r

Interference, superposition

Q: What happens when two waves “collide?”

A: They ADD together! We say the waves are superposed.

Constructive Destructive

Why superposition works

The wave equation is linear:

If f1 and f2 are solution, then Bf1 + Cf2 is also a

solution!

These points are nowdisplaced by bothwaves

2 2

2 2 2

10

y yx v t

(It has no terms where variables are squared.)

Superposition of two identical harmonic waves out of phase

Two identical waves out of phase:

1 , cosy x t A kx t 2 , cosy x t A kx t

intermediateconstructive destructive

Wave 2 is little ahead or behind wave 1

Superposition of two identical harmonic waves out of phase: the math

1 , cosy x t A kx t 2 , cosy x t A kx t

1 2, , , cos cos

cos cos

y x t y x t y x t A kx t A kx t

A kx t kx t

cos cos 2cos cos2 2

a b a ba b

, 2 cos cos2 2

y x t A kx t

When interf erence is completely destructive

0 interf erence is completely constructive

It’s all about the phase difference

Interference for 2D and 3D waves

Let S1, S2 be two sources that emit spherical sound waves in phase.

S1

S2

P

d1

d2

At point P:

2 1 2 1k d d nDestructive interference

2 1Phase diff erence k d d

This is what matters…

2 1 2 12

d d n

At some points the resulting wave can be very faint!

0,1,2...n

These points are always nodes!

Reflected waves: fixed end.

A pulse travels through a rope towards the end that is tied to a hook in the wall (ie, fixed end)

Fon wall by string

Fon string by wall

The force by the wall always pulls in the direction opposite to the pulse.

Another way (more mathematical): Consider one wave going into the wall and another coming out of the wall. The superposition must give 0 at the wall. Virtual wave must be inverted:

The pulse is inverted (simply because of Newton’s 3rd law!)

Reflected waves: free end.

A pulse travels through a rope towards the end that is tied to a ring that can slide up and down without friction along a vertical pole (ie, free end)No force exerted on the free end, it just keeps

going

Fixed boundary condition

Free boundary condition

Reflection on fixed end – inversion, or change of phase of π

Reflection on free end – no inversion

Standing waves

A wave traveling along the +x direction is reflected at a fixed point. What is the result of the its superposition with the reflected wave?

( , ) cos( ) cos( )y x t A kx t kx t

Standing wave

( , ) 2 sin( )sin( )y x t A kx t

cos cos 2sin sin2 2

a b a ba b

Nodes

No motion for these points (nodes)

2k

( , ) 2 sin( )sin( )y x t A kx t

Antinodes

2k

( , ) 2 sin( )sin( )y x t A kx t

These points oscillate with the maximum possible amplitude (antinodes)

Standing waves and boundary conditions

We obtained

Nodes 0, , ,... 2

x

3Antinodes , ...

4 4x

We need fixed ends to be nodes and free ends to be antinodes!

Strong restriction on the waves that can “survive” with a given set of boundary conditions.

( , ) 2 sin( )sin( )y x t A kx t

Normal modes

Which standing waves can I have for a string of length L fixed at both ends?

I need nodes at x = 0 and x = L Nodes 0, , ,... 2

x

, ,... f or 1,2,...2 2

L n n

2 f or 1,2,...n

Ln

n

Allowed standing waves (normal modes) between two fixed ends

Mode n = n-th harmonic

DEMO: Normal modes

on string

Ruben’s tube