Lecture 6Wave energy.

Interference. Standing waves.

ACT: 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:

(a) increase

(b) decrease

(c) stay the same

v

Fv

Tension is greater near the top because it has to support the weight of rope under it!

is greater near the top.v

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

Transfer of energy

The string to the left of x does work on the string to the right of x, just as your hand did:

x

Energy is transferred or propagated.

Power

What is the work done on the red segment by the string to its left?

1 1 tany x

yF F F

x

and is subject to a force whose y component is

F1

F

F1y

θ

1with (tension in the string)xF F

x

y

y

t

The red segment moves in the y direction with velocity

1 1 1 1

Power (how does energy move along the wave)

y y

y yP F v F v P F

x t

Power for harmonic waves

For a harmonic wave,

2 2siny y

P F Fk A kx tx t

, cosy x t A kx t

sin and siny y

kA kx t A kx tx t

Power (energy flow along x direction)

NB: Always positive, as expected

2 2 2, sinP x t F A kx t

kv F

Maximum power where vertical velocity is largest (y = 0) 2sin kx t

cos kx t

Average power for harmonic waves

2 2 2sinP F A kx t Average over time:

2 2

2 0sin 1

sin2 2

d

2 2 2 21 12 2

vP F A A

Average power for

harmonic waves

It is generally true (for other wave shapes) that wave power is proportional to the speed of the wave v and its amplitude squared A2.

ACT: Waves and friction

Consider a traveling wave that loses energy to frictionIf it loses half the energy and its shape stays the same, what is amplitude of the wave ?

A. amplitude decreases by ½B. amplitude decreases by 1/√(2)C. amplitude decreases by ¼

Power is proportional the amplitude squared A 2.

Interference, superposition

Q: What happens when two waves “collide?”

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

“Constructive” “Destructive”

Aside: Why superposition works

The wave equation is linear: It has no terms where variables are squared.

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

These points are nowdisplaced by bothwaves

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

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)

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

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:

DEMO: Reflection

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

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?

No motion for these points (nodes)

2k

These points oscillate with the maximum possible amplitude (antinodes)

Standing wave

cos cos 2sin sin2 2

a b a ba b

+x

-x

Standing wave

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!

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

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

2 free ends

1 fixed, 1 free

2 free ends

1 fixed, 1 free

2 fixed ends

2 free ends

1 2L 2 L 3

23L

4 2L Normal modes for fixed

ends (lower row)

First harmonic

Second harmonic

…

Normal modes 2DNormal modes 2D

For circular fixed boundary

DEMO: Normal modes square surface