Practice Questions Flashcards
general transverse wave
y(x,t) = Acos{kx±ωt+φ}
where φ is the offset
k =
2 pi / λ
positive propogation
to the left with a negative sign
i.e. kx - ωt
negative propogation
to the right with a positive sign
i.e. kx + ωt
the offset, φ, can be determined via
the boundary conditions
phase change is
ωt
dy/dt =
v
dv/dt =
a
displacement corresponds to which component
the REAL component
max values of displacement when
cos{kx±ωt} = 1
{kx±ωt} = 2 pi n
Show y(x,t) is a solution to the wave equation
- take the derivatives of y(x,t)
- substitute the derivatives into the wave equation
- solve to find a known relation
Energy density =
E/δx
δx =
∫ dx
progressive transverse wave
y(x,t) = y(x-ct)
where u = x-ct
dy/dt for a progressive transverse wave =
dy/du du/dt
dy/dx for a progressive transverse wave =
dy/du du/dx
units of impedance Z
kgs^-1
to convert from g to kg
x10^-3
linear density
rho, p
Transmitted energy coefficient T =
Et/Ei = C^2/A^2 Z2/Z1 = 4Z1Z2/(Z1+Z2)^2
mass derivation thingy
- Draw a Diagram
- Evaluate yi + yr = yt at boundary conditions (A+B=C)
- F = ma
- Fnet = T dy(t)/dx - T (dy(i)/dx + dy(r)/dx)
- substitute to F = ma
- evaluate derivatives at Boundary conditions
- multiply by -i
- C = A + B
- kT = Zω
for progressive waves to not be reflected
use impedance matching
impedance matching
Z2 = √(Z1Z3)
and
L = λ/4
Reflected Energy Coefficient R =
Er/Ei = B^2/A^2 Z1/Z1 = (Z1-Z3/Z1+Z3)^2
if at t = 0 the string is in equilibrium then the Epot =
0
ms =
ρL
calculating the energy residing in a mode
- Standing wave y(x,t)
- find if A or B is zero for the boundary conditions
- simply original y(x,t)
- take the derivative and evaluate for boundary conditions
- make use of orthogonality of sin function
- integrate LHS and RHS separately
- Solve for An or Bn
- Substitute into En
E is proportional to
the velocity squared
plucked string diagram for L/2
see notes
released from rest at time t=0
displaced
- find the derivative of y(n) then apply boundary conditions
- rewrite y(n) based on 1. then again apply boundary conditions
Pressure
P = nRT/V
number of moles n
n = M/m0
where m0 is the molecular weight
density
p = M/V
speed in pistons
c = √(γP/p)
P is pressure
and p is density
longitudinal waves in a solid bar
wave equation
normal wave equation but replace c^2 for Y/p
where Y is youngs modulus and p is density
standing waves where one end is fixed and the other free
Ψ(x,t) = Asin(kx+θ)cos(ωt+φ)
at the free end of a thin bar
we have balanced forces
T = YA ∂Ψ/∂x = 0
k value for a string where one is fixed and one is free to move
k(n) =((2n-1)π )/ (2l)
where l/L is length
Elastic energy
E = 1/2 k Δl^2
where k is the spring constant
spring constant k
k = F/Δl
Youngs modulus Y
Y = (F/A) / (l/Δl)
A membrane is a
2D wave equation
general solution to the 2D wave equation
z = 2iAexp{i(ωt-k(x)x)} sin(nπy/b)
2iAexp{i(ωt-k(x)x) representing a wave travelling the positive x-direction at velocity c = ω/k(x)
sin(nπy/b) shape of propagation in y direction
cut off frequency requires kx
to be real
for kx to be real
kx > 0
group velocity
v(g) = ∂ω/∂k
how long does it take a wave burst to travel
t = x/v(g)
phase velocity
v(p) = ω/k
and v(p) = c^2/v(g)
if a membrane is clamped round all edges
z = 0
and hence
k(x) = nπ/a and k(y) = nπ/b
dispersion equation
ω = 2 √(K/m) sin(ka/2)
where K the spring constant = YA/a
for max sin(ka/2) = 1
amu units
measured in gmol^-1 amu
spring constant units
Nm^-1
the wavefunction for a slit of finite width is
Ψ(x) = Aasinc(k(x)a/2)
Intensity can be found via
|Ψ(x)|^2
The diffraction pattern from the infinitely thin triple slit apparatus
Ψ(x) = Ao{1+2cos(k(x)d)}
Infinitely narrow double-slit problem , double slit
Ψ(x) = Ao cos(k(x)a/2)
