Equation of a wave
𝑦(𝑥, 𝑡) = ym sin⁡(kx − 𝑤t + ϕ)
Amplitude
the amplitude (ym) is the maximum displacement of a particle.
Remember partials in a wave are only displaced vertically.
Measured in metres.
Wave length
wavelength (λ)⁡is⁡the⁡distance⁡between⁡repetition of the wave (eg the distance between peaks, or
troughs or any points in phase)
Measured in metres.
Period
The Period (T) is the time between repetitions of the wave.
It is the time it takes for a particle to go from it’s max displacement (in one direction) back through
to it’s max displacement (in that same direction) again.
Measured in seconds.
Angular Wave number
The angular wave number (k) is defined by 𝑘 =
2𝜋
𝑟𝑎𝑑𝑖𝑎𝑛𝑠/𝑚.
𝜆
Where λ is the wave length.
Angular Frequency.
w is the angular frequency of the wave 𝑤 =
2𝜋
𝑇
= 2𝜋𝑓⁡𝑟𝑎𝑑𝑖𝑎𝑛𝑠/𝑠.
Frequency
The frequency (f) is how many wave cycles per second.
1
𝑤
Defined by 𝑓 = 𝑇 = 2𝜋
Measured in Hertz (cycles/second)
Phase constant
Adding a phase constant (ϕ) moves the wave ϕ units to the left, but it still keeps it’s shape.
also phase difference, phase angle.
Phasor
vector. Length = amplitiude, angle= phase difference (relative to some standard wave). Can be used
to add waves
Wave speed
𝑑𝑥
The waves speed is v= 𝑑𝑡 =
𝑤
𝑘
𝜆
= 𝑇 = 𝜆𝑓
Transverse speed
𝜕𝑦
The waves transverse speed (u) is the speed that the particles move vertically. u= 𝜕𝑡 =
−𝑤⁡ym cos⁡(kx − 𝑤t)
x must be a constant (doesn’t matter what) to find the transverse speed, since we only want to look
at one point on the wave, can how it moves up and down. As x is a constant it is a partial
𝜕𝑦
derivative⁡ 𝜕𝑡 .
Transverse Acceleration
Like all Simple Harmonic Motion (SHM) equations.
𝑎𝑦 = −𝑤 2 𝑦
Wave Speed in a stretched string.
wave speed in a string in determined by the linear density of the string 𝜇 =
𝑚
𝑙
kg/m.
And the tension τ.
𝜏 𝜆
𝑣=√ =
𝜇 𝑓
Is v transverse or longitudinal speed?
When the speed increases the wave length increases.
But increasing the wavelength (by decreasing the frequency) will not increase the speed (as the
speed is set only by the linear density and the tension).
When the speed increases the frequency remains the same because the frequency is set only by that
which generates the wave.
Energy of a Wave travelling a string
kinetic Energy
KE∝u (u is transverse velocity)
u=
𝑑𝑦
𝑑𝑡
At max amplitude (y=±ym) is a global maximum/minimum
thus
𝑑𝑦
𝑑𝑡
= 0 ⇛ 𝑢 = 0 ⇛ 𝐾𝐸 = 0
At peaks we have zero kinetic energy
𝑑𝑦
at y=0, is a point of infliction 𝑑𝑡 is at max
𝑑𝑦
𝑎𝑡⁡𝑚𝑎𝑥 ⇛ 𝑢⁡𝑎𝑡⁡𝑚𝑎𝑥 ⇛ 𝐾𝐸𝑎𝑡⁡𝑚𝑎𝑥
𝑑𝑡
at y=0 we have maximum Kinetic energy (a particle it moving at it’s fastest (up and down))
Like all Simple harmonic moth
Elastic potential energy
As can be seen in the diagram the string element dx, has maximum stretch at y=0.
and minimum (in fact zero) at y=ym.
stretch ∝EPE (elastic potential energy). So at y=0 there is maximum EPE, and at y=ym there is zero
EPE.
Waves transporting energy
as can be clearly seen (ith a little thought) waves transfere energy in the direction they propergate.
The areas with max energy moves in the direction of propergation ans time increases.
Rate of energy transfer (power)
1
2
𝑃𝑎𝑣𝑔 = 𝜇𝑣𝑤 2 𝑦𝑚
⁡
2
μ⁡&⁡v⁡depend⁡on⁡the⁡tension⁡and⁡linear⁡density⁡of⁡the⁡sting
𝑤⁡&⁡𝑦𝑚 ⁡𝑑𝑒𝑝𝑒𝑛𝑑⁡𝑜𝑛⁡𝑡ℎ𝑒⁡𝑤𝑎𝑣𝑒′𝑠⁡𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑡ℎ𝑎𝑡⁡𝑖𝑠⁡𝑤ℎ𝑎𝑡⁡𝑤𝑎𝑠⁡𝑢𝑠𝑒𝑑⁡𝑡𝑜⁡𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒⁡𝑡ℎ𝑒⁡𝑤𝑎𝑣𝑒⁡⁡⁡
Is v transverse or longitudinal speed?
Wave Equation
𝛿 2𝑦
1 𝛿 2𝑦
=
𝛿𝑥 2 𝑣 2 𝛿𝑥𝑡 2
What is it good for?
Interference
we have 2 waves:
𝑦1(𝑥,𝑡) = ym sin(kx − 𝑤t)
𝑦2(𝑥,𝑡) = ym sin(kx − 𝑤t + ϕ)
The same but with phase difference ϕ radians.
The result is a (moving) sinusoidal wave with different amplitude (as defined above), but the same
wavelength, and frequency.
𝑦1(𝑥,𝑡) = ym sin(kx − 𝑤t)
⁡⁡⁡⁡⁡⁡⁡⁡𝑦2(𝑥,𝑡) = ym sin(kx − 𝑤t + ϕ)
Phase Difference, in
Degrees Radians Wavelengths
0
0
Amplitude of Resultant Wave Type of Interference
0
2ym
Fully constructive
120
2π/3
0.33
ym
Intermediate
180
π
0.50
0
Fully destructive
240
4π/3
0.67
ym
Intermediate
360
2π
1.00
2ym
Fully constructive
865
15.1
2.40
0.60ym
Intermediate
Standing waves
When two identical waves heading towards each other interfere the result is a standing wave.
a standing wave does not move along the x axis as time increases instead, it have point that don’t
move - (displacement) nodes, and between them as time increases the other points oscillate
transversely. The points that oscillate to most are called antinodes.
Standing waves are define at time t, and position x, to have displacement y(x,t) as follows:
𝑦(𝑥, 𝑡) = (2ym sin kx) cos wt⁡⁡
Not all points in a standing wave have the same max amplitude.
Instead it varies with the point/position. The nodes always have zero amplitude.
The antinodes have the greatest amplitude.
The amplitude of a point is defined (at position x) by 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = (2ym sin kx)\
Nodes
Nodes are located every half wavelength (that is at each x intercept, for obvious reasons).
Therefore their position in defined by the equation:
𝜆
𝑥=𝑛 ,
𝑛 = (0,1,2,3 … )
2
Antinodes are located halfway between them.,thus their equation is
𝜆 𝜆
𝑥=𝑛 + ,
𝑛 = (0,1,2,3 … )
2 4
1 𝜆
𝑥 = (𝑛 + ) ,
𝑛 = (0,1,2,3 … )
2 2
Reflection
(a) A pulse incident from the right is reflected at the left end of the
string, which is tied to a wall. Note that the reflected pulse is
inverted from the incident pulse. (b) Here the left end of the string
is tied to a ring that can slide without friction up and down the
rod. Now the pulse is not inverted by the reflection.
.
QUOTE FoP page 432
In Fig. 16-21a, the string is fixed at its left end. When the pulse arrives at that end, it exerts an
upward force on the support (the wall). By Newton’s third law, the support exerts an opposite force
of equal magnitude on the string. This second force generates a pulse at the support, which travels
back along the string in the direction opposite that of the incident pulse. In a “hard” reflection of this
kind, there must be a node at the support because the string is fixed there. The reflected and
incident pulses must have opposite signs, so as to cancel each other at that point.
In Fig. 16-21b, the left end of the string is fastened to a light ring that is free to slide without friction
along a rod. When the incident pulse arrives, the ring moves up the rod. As the ring moves, it pulls
on the string, stretching the string and producing a reflected pulse with the same sign and amplitude
as the incident pulse. Thus, in such a “soft” reflection, the incident and reflected pulses reinforce
each other, creating an antinode at the end of the string; the maximum displacement of the ring is
twice the amplitude of either of these pulses.
QUOTE FoP page 432
So fixed ends always have a node.
Free ends have an antinode. (will reflection cause standing waves if there is a free end?, can
resonance occur?)
Reflected waves can form standing waves. They will reinforce the original themselves
Resonance
For certain frequencies the reflected wave reinforces it’s the original wave (forming a standing
wave), the wave is then reflected again and reinforces again (forming a larger standing wave).
Without this a guitar sting for example would make very little sound.
The frequency at which the wave has to be (to set up a standing wave) is called is called it’s resonant
frequency. A string has multiple resonant frequencies.
Since both ends of the string are fixed there must be a node at each end.
So we can have a antinode halfway between them. ∴ λ⁡=2L⁡⁡so⁡L=λ/2
-(a)
or we⁡could⁡have⁡a⁡node⁡there⁡and⁡antinodes⁡⁡between⁡them⁡∴ λ⁡=L⁡⁡so⁡L=2λ/2= λ -(b)
or⁡we⁡could⁡have⁡a⁡nodes⁡there⁡and⁡antinodes⁡⁡between⁡them⁡∴ λ⁡=2L/3⁡⁡so⁡⁡L=3λ/2 –(c)
Thus for string of length L,
The resonant wavelengths are:
𝜆=
And so the resonant frequencies are 𝑓 =
2L
,⁡⁡⁡⁡⁡⁡⁡for⁡n = (1,2,3 … )
n
v
𝑛 2L ,⁡⁡⁡⁡for⁡n = (1,2,3 … )
All the resonant frequencies are integers multiples of the
fundamental frequency (that is the frequency when n=1)
Call the resonant frequencies harmonics, the fundamental
frequency is the first harmonic, the next one (when n=2) is the
second harmonic, and so on – the nth harmonic.
We name the harmonics 𝑓1 , 𝑓2 , 𝑓3 , 𝑓𝑛
the harmonic number (n) is equal to the number of antinodes, (number of ‘bubbles’)
Resonance can damage buildings, make things shake etc etc