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General Details • 12 lectures
• Notes will be available to download every 1-2 weeks.
• These will NOT be complete, so pay attention in the lectures
• Unintentional mistakes may occur – please let me know in lectures or via email if you spot anything
• 3 problem sheets will be distributed. These are inherited from previous lecturers of this course – many thanks to them
• Material will be posted on https://www2.physics.ox.ac.uk/contacts/people/jarvis (under “Teaching”)
• Thanks also to Guy Wilkinson for his lecture notes on this course
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Text Books
• Vibrations & Waves, A.P. French, MIT Introductory Physics Series
• Vibrations and waves in Physics, I.G. Main, CUP
• Waves, C.A. Coulson & A. Jeffrey, Longman Mathematical Texts
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VibraIons & Waves in Physics
• Although we will only use specific examples during this course, the physics of waves and vibrations underpin many different areas of physics that you will comes across over the next 3-4 years.
• For example: • Electromagnetism • Quantum Mechanics • Cosmology
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Coupled Pendula
EquaIons of MoIon for each pendulum…
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How to solve? Equations of Motion: Two equations and two unknowns, x and y First we will use the so-called `decoupling method’ Which involves decoupling the equations from each other and solving the decoupled equations individually.
(1)
(2)
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Decoupling Method Equations of Motion: Adding (1) and (2)
(1)
(2)
Looks very similar to the standard wave equation… define
with
SHM!!! A1 & ϕ1 are constants set by boundary conditions
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Decoupling Method Equations of Motion: Subtracting (2) from (1)
(1)
(2)
This time
with
A2 & ϕ2 are constants set by boundary conditions
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Normal Modes of Coupled Pendulum • The first normal mode: centre-of-mass motion
• The second normal mode: relative motion
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Normal Modes of Coupled Pendulum The variables q1 and q2 are called the mode, or normal, coordinates In any normal mode only one of these coordinates is active at any one time (i.e. either q1 is vibrating harmonically and q2 is zero or vice versa) It is more common to define the mode coordinates with a normalising factor in front (in this case 1/√2 )
This means that the vector defined by (q1,q2) has same length as that defined by (x,y), i.e. q12 + q22 = x2 + y2. This factor changes none of results we obtained.
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General SoluIon for a Coupled Pendulum
The General Solution is a sum of the two normal modes
The constants are just set by the initial conditions
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Coupled Pendulum: Different IniIal CondiIons
Example 1 Plugging these iniIal condiIons into the general soluIon gives:
x
-y
x
+y
y
x
Time
a
-a
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Example 2 Gives…
x-y
x+y
y
x
Time
-v/ω2
v/ω2
Coupled Pendulum: Different IniIal CondiIons
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Example 3 Gives…
x-y
x+y
y
x
Time
-a
a
Coupled Pendulum: Different IniIal CondiIons
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Example 3
Coupled Pendulum: Different IniIal CondiIons
Let…
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Example 3
x
-y
x
+y
y
x
Time
-a
a In this example both of the normal modes are excited
‘Beats’ – energy is being transferred between pendula
Coupled Pendulum: Different IniIal CondiIons
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Coupled Pendulum: Energy
Calculate total energy of the system U = KE + PE = T + V
Kinetic Energy
Potential Energy
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Coupled Pendulum: Energy
Can also calculate Potential Energy using
Neglecting the constant, C, which is an arbitrary offset
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Coupled Pendulum: Energy
This is a bit unwieldy. Why don’t we go back to the normal coordinates and see what it looks like?
The cross-term in V has now disappeared
Energy in mode 1 Energy in mode 2
Total energy in the system = sum of energies in each mode
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With –w being the eigenvalues
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Unequal coupled pendula
• Up unIl now we have only considered the case where the 2 pendula were of the same length
• Now we will find the equaIons of moIon for pendula of unequal length
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