PHY 102: Waves & Quanta

Topic 2

Travelling Waves

John Cockburn (j.cockburn@... Room E15)

•What is a wave?

•Mathematical description of travelling pulses & waves

•The wave equation

•Speed of transverse waves on a string

TRANSVERSEWAVE

LONGITUDINALWAVE

WATER WAVE(Long + TransCombined)

•Disturbance moves (propagates) with velocity v (wave speed)

•The wave speed is not the same as the speed with which the particles in the medium move

•TRANSVERSE WAVE: particle motion perpendicular to direction of wave propagation

•LONGITUDINAL WAVE: particle motion parallel/antiparallel to direction of propagation

No net motion of particles of medium from one region to another: WAVES TRANSPORT ENERGY NOT MATTER

Mathematical description of a wave pulse

-8 -6 -4 -2 0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

y

X

f(x) f(x-10)f(x+5)GCSE(?) maths:

Translation of f(x) by a distance d tothe rightf(x-d)

0.0

0.2

0.4

0.6

0.8

1.0

y

X 0

d=vt

For wave pulse travelling to the right with velocity v :

f(x) f(x-vt) )(),( vtxftxf

function shown is actually:2)(),( vtxetxf

Sinusoidal waves

Periodic sinoisoidal wave produced by excitation oscillating with SHM (transverse or longitudinal)

Every particle in the medium oscillates with SHM with the same frequency and amplitude

Wavelength λ

Sinusoidal travelling waves: particle motion

Disturbance travels with velocity v

Travels distance λ in one time period T

fvT

v

vT

Sinusoidal travelling waves: Mathematical description

Imagine taking “snapshot” of wave at some time t (say t=0)

Dispacement of wave given by;

x

Atxy2

cos)0,(

If we “turn on” wave motion to the right with velocity v we have (see slide 5):

)(2cos),(

vtxAtxy

)(2cos),(

vtxAtxy

Sinusoidal travelling waves: Mathematical description

We can define a new quantity called the “wave number”, k = 2/λ

)cos(),( kvtkxAtxy

)cos(),(

2

tkxAtxy

kk

ffv

NB in wave motion, y is a function of both x and t

The Wave Equation Curvature of string is a maximumParticle acceleration (SHM) is a maximum

Curvature of string is zeroParticle acceleration (SHM) is zero

So, lets make a guess that string curvature particle acceleration at that point……

The Wave Equation

Mathematically, the string curvature is:2

2 ),(

x

txy

And the particle acceleration is:2

2 ),(

t

txy

So we’re suggesting that: 2

2

2

2 ),(),(

t

txy

x

txy

)cos(),( tkxAtxy

The Wave Equation

)sin(),(

tkxkAx

txy

)cos(),( 2

2

2

tkxAkx

txy

)sin(),(

tkxAt

txy

)cos(),( 2

2

2

tkxAt

txy

2

2

22

2 ),(1),(

t

txy

vx

txy

Applies to ALL wave motion (not just sinusoidal waves on strings)

Wave Speed on a string

Small element of string (undisturbed length ∆x) undergoes transverse motion, driven by difference in the y-components of tension at each end (x-components equal and opposite)

T2

T1

y

x

T

T

T2y

T1y

x+∆x

Small elementof string

∆x

motion

Wave Speed on a string Net force in y-direction:

yyy TTF 12

T2y, T1y given by:

2

2

t

y

x

xy

Txy

Txxx

From Newton 2, :

2

2

2

2

dt

yx

dt

ymFy

xy

xxy x

yTT

x

yTT

12 ;

Wave Speed on a string

2

2

x

yT

x

xy

Txy

Txxx

Now in the limit as ∆x0:

So Finally:

2

2

2

2

t

y

Tx

y

Comparing with wave equation:

2

2

22

2 ),(1),(

t

txy

vx

txy

T

v