phy 102: waves & quanta topic 2 travelling waves john cockburn (j.cockburn@... room e15)
DESCRIPTION
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. TRANSVERSE WAVE. LONGITUDINAL WAVE. WATER WAVE (Long + Trans - PowerPoint PPT PresentationTRANSCRIPT
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