Waves Flashcards
what is a wave
a collective bulk disturbance that propagates through a medium, in which whatever happens at any specific point is a delayed response to the disturbance at adjacent points
the wave carries only
energy as it has no net displacement
you get a wave if and only if
there is finite speed of propagation of the forces that move the disturbance
Transverse waves on a string
wiggle one end and a pulse moves along the string
what determines the propagation speed of the pulse
tension in the string and the string’s mass density
the wave speed will increase if
the forces/tension are higher
or
the mass density is lower
c² ∝
T/ρ
The element does not move along the string, only moves
transversely as pulse propogates
The wave equation
derived from the net force of the transverse forces = ma
δ²y/δx²= ρ/T * δ²y/δt²
superposition
if we have solutions y₁(x,t) and y₂(x,t) then any combination Ay₁(x,t) ± By₂(x,t) will also be a solution
The kinetic energy in a travelling wave
Ek = 1/2δmv^2 = 1/2 ρ *δx *(δy/δt)²
The potential energy in a travelling wave
Ep = Tδx/2 (δy/δx)²
δy/δt = dy/du =
-c δy/δx
Total energy for once cycle
E = 1/2ρA²ω²λ
Power formula
P = 1/2A²ω²√Tρ
boundary
- free to move
restoring force is zero
reflected wave has same polarity (no phase change) as incident wave
boundary
- fixed
displacement is zero
reflected wave has inverted polarity
counter-propagating waves definition
sine waves travelling in opposite directions create standing waves when they meet
each element of the string appears to execute - synchronously - the same transverse motion
counter-propagating waves mathematically described
y(x,t) = X(x)T(t)
A standing wave is equivalent to
the sum of two counter-propagating travelling waves of the same frequency and amplitude
energy densities
energy density = ρ/2(∂y/∂t)^2 + T/2 (∂y/∂t)^2
the energy divided by the unit length δx.
deriving the wave equation
F = ma
T [∂y/∂x |xo+∂x/2 - ∂y/∂x | xo - ∂x/2 ] = ρdx ∂^2y/∂t^2 | xo
T ∂^2y/∂x^2 | xo = ρ ∂^2y/∂t^2
how to derive kinetic Eone cycle
take the transverse wave y(x) = A cos(kx-wt)
K = 1/2 (λ ∫ 0) (dy/dt)^2 p x dx
similar for U.
