1 Waves 4
Lecture 4 - Stretched stringLecture 4 - Stretched string
Transverse waves on a stretched stringTransverse waves on a stretched string
Aims:Aims: Derivation of the wave equation:
obtain the wave velocity, etc. Wave impedance:
characteristic impedance of a stretched string.
Polarization. Linear; circular; Polarization and coherence.
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General approach:General approach: Consider a small segment; Find the difference in forces on the two ends; Find the element’s response to this imbalance.
We need the following useful result:We need the following useful result:
easy to see graphically:
Waves on a string:Waves on a string: Neglect extension of the string i.e. tension is
constant everywhere and unaltered by the wave. Displacements are small so
<<. x and are << 1.
Neglect gravity
Derivation of the wave equationDerivation of the wave equation
xdxdg
xgxxg
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Stretched stringStretched string
Forces on an element of the string:Forces on an element of the string:
Net force in the y-direction is
But
so
Net transverse force =
To apply Newton’s 2nd law we need mass and acceleration. These are:
mass of element =
acceleration (transverse) =
tantansinsin TTTT
xxxx
tan
xtan
xx
T
2
2
x22 t
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Derivation of wave equationDerivation of wave equation
Newton’s second law:Newton’s second law:
The wave velocity is not the transverse velocity of the string, which is
Two Polarization's Two Polarization's for a transverse wave: Horizontal (z) ; Vertical (y).
2
2
2
2
txx
xT
T
tTx velocity wave
2
2
2
2
massmass accelerationaccelerationforceforce
.t
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Wave impedanceWave impedance
The general concept:The general concept: Applying a force to a wave medium results in a
response and we can therefore define an impedance = applied force/velocity response.
Expect the impedance to be real (force and velocity are in-phase) - since energy fed into the medium propagates (without loss) away from the source of the excitation. Imaginary impedance - no energy can be
transported (examples later: waveguide below cut-off).
Complex impedance - lossy medium.
Characteristic impedance:Characteristic impedance:
Transverse driving force:
Transverse velocity:
xTTTF
tansin
t
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Impedance of a stretched stringImpedance of a stretched string
ImpedanceImpedance
For a wave in +ve x-direction, recall
Impedance (for wave in +ve x-direction)
Wave in -ve x-direction, Z has the opposite sign:
Note that the impedance is real. i.e. the medium is lossless (in this idealised picture).
txT
Z
velocity transverseforce driving
tvx
tf
vtu
fx
1
;
so,
vTvTZ
vTvTZ
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Energy in a travelling waveEnergy in a travelling wave
Energy is both kinetic and potential.Energy is both kinetic and potential. Kinetic energy associated with the velocity of
elements on the string. Potential energy associated with the elastic
energy of stretching during the motion.
Potential energy density Potential energy density (P.E./unit length) Calculate the work done increasing the length
of segment x against a constant tension T.
Increase in length of segment is
xx
xx
x
xxx
x
22
2122
21
21
1
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Energy densityEnergy density
Increase in P.E Increase in P.E (= force x extension)
Potential energy densityPotential energy density
Kinetic energy densityKinetic energy density K.E. of length x
Density
Note: instantaneous KE and PE are equal since
Total energy densityTotal energy density
Sum of KE and PE
xx
T
2
21.
extensionextensionforceforce
2
21.
x
T
2
21
t
x
1/21/2 mm v2v2
2
21
t
21 vTtvx and
2
t
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Energy: harmonic waveEnergy: harmonic wave
Average energy densityAverage energy density (section 1.1.3)
average energy density is
consider each element, length x, as an oscillator with energy
In time t we excite a length x=vt. So the energy input is
Since Z=v,
Mean powerMean power
2; okxti
o AeA
tt21
221
oA
221
oAx
221
oAtv
2221
oAZ