Review: Waves - I

Waves

Particle: a tiny concentration of matter, can transmit energy.

Wave: broad distribution of energy, filling the space through which it travels.

Quantum Mechanics:

Wave Particle

Types of Waves

Types of waves: Mechanical Waves, Electromagnetic Waves, Matter Waves, Electron, Neutron, People, etc ……

Transverse Waves:

Displacement of medium Wave travel direction

Longitudinal Waves:

Displacement of medium || Wave travel direction

Parameters of a Periodic Wave

: Wavelength, length of one complete wave form

T: Period, time taken for one wavelength of wave to pass a fixed point

v: Wave speed, with which the wave moves

f: Frequency, number of periods per second

= vT v = T = f

Wave Function of Sinusoidal Waves

y(x,t) = ymsin(kx-t)

ym: amplitude

kx-t : phase

k: wave number k 2

When ∆x=, 2 is added to the phase

: angular frequency 2T2f

When ∆t=T, 2 is added to the phase

Wave SpeedHow fast does the wave form travel?

Wave Speed

How fast does the wave form travel?

Pick a fixed displacement a fixed phase

kx-t = constant v dx

dtk

y(x,t) = ymsin(kx-t) v>0

y(x,t) = ymsin(kx+t) v<0

v Transverse Waves (String):

Principle of Superposition

Overlapping waves add to produce a resultant wave

y’(x,t) = y1 (x,t) + y2 (x,t)

Overlapping waves do not alter the travel of each other

Interference

y t y1 y2 2ym cos1

2

sin kx t

1

2

y1 t ym sin kx t y2 t ym sin kx t

n=0,1,2, ...

Constructive:

Destructive:

k 2 n

n 1

2

Phasor Addition

PHASOR: a vector with the amplitude ym of the wave and rotates around origin with of the wave

When the interfering waves have the

same PHASOR ADDITIONINTERFERENCE

Can deal with waves with different

amplitudes

Standing Waves

Two sinusoidal waves with same AMPLITUDE and WAVELENGTH traveling in OPPOSITE DIRECTIONS interfere to produce a standing wave

y x, t y1 y2 2ym sinkx cost

sin sin 2sin1

2 cos

1

2

The wave does not travel

Amplitude depends on position

1 , sinmy x t y kx t 2 , sinmy x t y kx t

y x, t 2ym sinkx cost

NODES: points of zero amplitude

kx n , or xn2

n0,1,2,...

ANTINODES: points of maximum (2ym) amplitude

kx n1

2

, or x n

1

2

2

n 0,1,2,...

k 2 sin n 0 sin n

1

2

1

Standing Waves in a String

The BOUNDARY CONDITIONS determines how the wave is reflected.

Fixed End: y = 0, a node at the end

Free End: an antinode at the end

The reflected wave has an opposite sign

The reflected wave has the same sign

Case: Both Ends Fixed

y x, t 2ym sinkx cost

y x 0 0 y x L 0

sin kL 0 k nL

, n1,2,3,....

k can only take these values

k 2

2L

nOR

f v

f nv

2LOR v where

RESONANT FREQUENCIES:

f n

2L

(a) k = 60 cm-1, T=0.2 s, zm=3.0 mm

z(y,t)=zmsin(ky-t)

= 2/T = 2/0.2 s =10s-1

z(y, t)=(3.0mm)sin[(60 cm-1)y -(10s-1)t]

uz z(y, t)

tzm cos ky t

zm sin2 (ky t)

(b) Speed

uz,min= zm = 94 mm/s

HRW 11E (5th ed.). (a) Write an expression describing a sinusoidal transverse wave traveling on a cord in the y direction with an angular wave number of 60 cm-1, a period of 0.20 s, and an amplitude of 3.0 mm. Take the transverse direction to be the z direction. (b) What is the maximum transverse speed of a point on the cord?

f = 500Hz, v=350 mm/s

x, t kx t(a) Phase

x, t 2f

vx 2ft

k 2

v f k

2f

2f

vx

x v

2f

350m/s

2 500Hz 30.117 m

(b) 2ft 2 500 Hz (1.0010 3 ) rad.

HRW 16P (5th ed.). A sinusoidal wave of frequency 500 Hz has a velocity of 350 m/s. (a) How far apart are two points that differ in phase by /3 rad? (b) What is the phase difference between two displacements at a certain point at times 1.00 ms apart?

y(x,t) = ymsin(kx-t)

y t 2ym cos1

2

sin kx t

1

2

y1 t ym sin kx t y2 t ym sin kx t

2

For

A2ym cos1

2 2ym cos

41.4ym

HRW 36E (5th ed.). Two identical traveling waves, moving in the same direction, are out of phase by /2 rad. What is the amplitude of the resultant wave in terms of the common amplitude ym of the two combining waves?

ym ym12 ym2

2 2ym1ym2 cos

ym12 ym2

2 2ym1ym2 cos4.4mm

(a)

h

ym2sin sin

h

ymsin (b)

sin ym2 sin

ym0.935

The angle is either 68˚ or 112˚. Choose 112˚, since >90˚.

HRW 41E (5th ed.). Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string with amplitudes of 4.0 and 7.0 mm and phase constant of 0 and 0.8 rad, respectively. What are (a) the amplitude and (b) the phase constant of the resultant wave?

ym1=4.0 mm

ym2=7.0 mm

ymh