String as Wave Guide

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String as Wave Guide
Acoustics of Music Semester 2

• Aims
• To introduce the concept of a string as medium
  for
• (i) wave propagation
• (ii) standing waves and modes of vibration
• Learning Outcomes
• Should be able to understand and derive equation
  of motion for string
• Should be familiar with key methods for solving
  this second order differential equation and their
  physical importance

2
Review of Last Week
We studied a string as two springs

• Discussed scenarios
• No tension
• Constant tension T
• Tension from extending springs of free length x0

To show linear and non linear behaviours
depending on tension and displacement - hence how
this relates to musical expression
linear
non-linear
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Consider Stretched String

• Assumptions
• uniform linear density
• negligible bending stiffness
• constant tension
• T large enough to rule out effects of gravity
• no dissipative forces
• small angles

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Consider Stiffness and InertiaLets get equation
of motion Fma
Net vertical tension
Mass per unit length
Mass of section of string
Assuming small angles
Hence
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For infinitesimally small section
Hence
Gives wave equation
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Wave Equation

• Were assuming tension and mass per unit length
  constant and hence the speed of wave constant

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Solving

• D'alemberts solution

Method of separation of variables
(Bernoulli)
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Some math revision (chain rule)

• If
• Then

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D'Alembert Method

• Introduce new variables u and v
• Hence we assume a solution of the form.
• yf(u(x,t),v(x,t))
• Essentially we need to differentiate to
  substitute in wave equation

10

• Consider
• Consider

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• Second Derivatives
• Similarly

12

• Hence our wave equation
• Becomes
• Which simplifies to

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Can write wave equation as
i.e. a constant with respect to u
Integrate with respect to u gives
Integrating again
In other words yarbitrary functions
D'Alemberts Solution
i.e.
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Travelling Waves

• two travelling waves going from left to right and
  right to left
• At a fixed end, say...
• Reflection at boundaries

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Method of separation of variables
Starting with assumed solution
Partial Differentiation
Substituting both into Wave Equation
Gives
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Collecting terms
Since left depends on t only, and right depends
on x only, their common value must be a constant.
hence
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Lets write
as
Hence
This gives standard solution
where C and D constants
Similarly
gives solution
where A and B are constants
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Solution to Wave Equation(Bernoulli - Separation
of Variables)
What does this tell us?
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Apply boundary conditions
String fixed at bridge and neck.
y(0,t)0
y(L,t)0
f(t) does care what x is so can remove f(t)
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Characteristic (Eigen) Values
Boundary Conditions
Only satisfied if C0 because of cos1 if x0.
If D0 no vibration
Consider
However equation satisfied if
Hence discrete set of frequencies
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Modes of Vibration
Collectively
oscillation term
restrictor
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Natural Frequencies of a String