TAP 222-4: Momentum questions
These questions change in difficulty and ask you to relate impulse to change of momentum.
1.
Thrust SSC is a supersonic car powered by 2 jet engines giving a total thrust of
180 kN.
Calculate the impulse applied to the car wh
Waveguide
Innitely long rectangular membrane, width b, xed at edges.
Displacement (x, y, t) satises 2-D wave equation
2
=
2 2
1 2
+
= 2 2
x 2 y 2
c t
Subject to (x, 0, t) = (x, b, t) = 0
Separate using (x, y, t) = X (x)Y (y)T (t) giving
d 2X
2
= kx X
dx
Outline
Convolution
Examples
Convolution Theorem
Convolution
Denition: The convolution of two functions f1 (x) and f2 (x) is
dened as
f1 (x u)f2 (u) du
F (x) =
Symmetric in f1 and f2 .
Example:
Convolution Example 1
Calculate the convolution
g( t)f (t) dt
Outline
Two more examples of PDEs and their solution
Laplaces Equation
Heat Flow Equation
Laplaces Equation
Electric eld: E = where is the electrostatic potential.
Gausss law: E = 0 - no enclosed charge.
Combine 2 = 0 - Laplaces equation
Problem/Example:
Outline
1-D Wave Problem: Waves on a string
Eigenvalues & Normal Modes
Superposition
Initial conditions
Wave Problems in 1D
A stretched string
What are these people doing (mathematically) ?
A stretched string
x=L
x=0
(x, t)
Transverse displacement (x, t)
Outline
Waves and Wave Packets
Wave Equation
2
x 2
=
1 2
c 2 t 2
where c is a constant.
Trial solution on < x <
(x, t) = ei(kxt) = eikx eit
k 2 = 2 /c 2 so (k) = ck and phase velocity vp = (k)/k = c.
Travelling waves
A plane wave of wavenumber k and angu
Outline
Fourier Transforms
The Dirac Function
Fourier Transform
Fourier Transform of a non-perodic function, f (x). If g(k) is the
FT of f (x), then
g(k) =
1
2
f (x) =
f (x)eikx dx
g(k)eikx dk
and f (x) is the inverse transform of g(k).
x and k have inver
Corrections and Clarications
In the previous lecture while working on the board and
substituting the exponential trail solution into the equation for
simple harmonic motion, a factor of i was missed.
The auxillary equation is 2 = k 2 so = ik.
Outline
Four
PHYS 20171 MATHEMATICS OF WAVES AND FIELDS
Lecturer: Gary Fuller1
Room: Turing Building 3.111
Email: [email protected]
Web page linked from Teachweb.
1
Also Year Tutor for 2nd Year & Careers Contact
Some example partial differential equations (PDE
Problems in 2 and 3 dimensions
Rectangular membrane xed to rectangular frame.
y
y =b
0
x =a x
Displacement (x, y, t) satises 2-D wave equation
2
=
2 2
+
x 2 y 2
=
1 2
c 2 t 2
1 2
c 2 t 2
Boundary conditions
(0, y , t) = (a, y , t) = (x, 0, t) = (x, b, t)