wave equation

A useful solution to the wave equation for an ideal string is:

 y(x,t) = Asin(2π/λ(x+/-vt))

This is a solution to the one-dimensional wave equation (via direct substitution):


∂ ² / ∂x² = (ρ/T)(∂²y/∂t²)

giving us:

1 =  (ρ/T)v²

giving the wave velocity of a stretched string:
v = √(T/ρ)
where T = tension in the string

and the mass her unit length:
ρ = m/L

Useful links

Useful resources:


http://www.physics.buffalo.edu/phy207/lc/lc15.pdf

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wavsol.html#c1



Identities to know:

     cos(x+λ) = cos(x)     and   λ = 2π
cos(x + λ/2) = -cos(x)   and   λ/2 = π
cos(x + λ/4) = -sin(x)   and   λ/4 = π/2


LOCATIONS OF THE WAVE ANGLE MAXIMA AND MINIMA
ΔL = dsin(θ)

MAXIMA:
ΔL = nλ

MINIMA:
ΔL = (n + 1/2)λ

The principle of superposition

From Mastering Physics:


The principle of superposition states:

If two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves y1(x,t)+y2(x,t), where y1(x,t) is the wave described in Part A and y2(x,t) is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

y_s(x,t) = y_e(x)y_t(t)


This form is significant because:

 y_e(x) called the envelope, depends only on position

 y_t(t) depends only on time


Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of ye(x).



COMBINING TWO WAVES TO CREATE A STANDING WAVE:

y1(x,t)=Asin(kx−ωt).

y2(x,t)    = Asin(kx+ωt)
y1(x,t)= 
  y1(x,t)=

Consider the sum of two waves y1(x,t)+y2(x,t)

Their sum can be written as follows:
ys(x,t)=ye(x)yt(t).

ye(x), yt(t) =

2Asin(kx),cos(ωt)

New topic ... Optics / waves and light!

MECHANICAL WAVES

General equations:

Amplitude in Simple Harminoc Motion (SHM): A = $\sqrt{\mathrm{Xi^2\; +\; (V^2/\omega \; ^2)}}$
 where ω = $\sqrt{\mathrm{k/m}}$

Phase angle (SHM): Φ = arctan(-Vi/ω xi)
 where ω = $\sqrt{\mathrm{k/m}}$$$$$

Phase: (kx-ωt) is a constant wavelength: λ = 2π/ω UNIT: meter


frequency: f = 2π/ω UNIT: cycles/second

NOTE THAT ω also equals: ω = νk 

wavespeed or speed of propagation: ν = $\sqrt{\mathrm{T/\mu }}$ UNIT: meters/second

ν = fλ

k = 2π/λ 

General Transverse Equation: y(x,t) = Acos2π([x/λ]-[t/T]) 

 Tension: μv = (mass of rod/length of rod)(frequency•wavelength)


POWER

Power: P = (force)(velocity) 

 P = 4πr^2(Intensity)

Converting power to energy? multiply by Joules! 1 W = 3.6x10^3 Joules