Oscillations and Waves Flashcards
Amplitude, A, and units
max displacement from eqlbm (max value of |x|)
units is metres
Period, T, and units
time it takes to complete one cycle - always positive
units is seconds per cycle
frequency, f, and units
number of cycles in a unit of time
unit is hertz
angular frequency eqn
ω = 2πf
= 2π/T
period is a reciprocal of…
frequency
Simple Harmonic Motion
when restoring force is directly proportional to the displacement of eqlbm
restoring force exerted by an ideal spring
F_x = -kx
k = force constant of spring
x = displacement
units of k are N/m or kg/s^2
equation for SHM displacement
x = Acos(ωt +φ)
A= amplitude
ω = angular frequency
t = time
φ= phase angle
x = displacement
displacement and acceleration always have…
opposite signs
acceleration eqn, a_x
a_x = -ω^2x
angular speed equation/ angular frequency for SHM
ω^2 = k/m
ω = sqrt(k/m)
frequency for SHM
f = 1/(2π)sqrt(m/k)
larger mass means acceleration is…
lower and it takes longer time to complete cycle
larger force constant k means…
it’s a stiffer spring and has a greater force, acceleration and has a quicker T per cycle
total mechanical energy in SHM eqn
E = (1/2)(mv^2) + (1/2)(kx^2)
= (1/2)(kA^2)
= constant
E = total mechanical energy
m = mass
v = velocity
k = force constant
x = displacement
A = amplitude
what’s v_x of an object at a given displacement x?
v_x = +/- sqrt(k/m) * sqrt(A^2-x^2)
k= force constant
m= mass
A= amplitude
x= displacement
whats v_max?
v_max = ωA
SHM w frictional damping force can be directly proportional to…
the velocity of the oscillating object
displacement of oscillator when damped
x = Ae^[-(b/(2m))t]cos(ω’t +φ)
x = displacement
A = amplitude
e = exponential
b = damped constant
m = mass
t= time
ω’ = angular frequency when damped
φ = phase angle
angular frequency when damped
ω’ = sqrt( k/m - b^2/(4m^2))
the larger the value of b…
the more quickly the amplitude decreases
when b = 2*sqrt(k/m)
critical damping so system no longer oscillates but returns to eqlbm position
when b>2*sqrt(k/m)
overdamping
when b < 2*sqrt(k/m)
underdamping
total mechanical energy for damped oscillations
E = (1/2)mx^2 + (1/2)kx^2
amplitude of a forced oscillator (damped)
A = F_max / sqrt [ (k-mw_d^2)^2 + b^2w_d^2]
F_max = max value of driving force
w_d = driving angular frequency
transverse wave
displacement of the medium is perpendicular to the direction of travel of the wave
longitudinal wave
particle movement is parallel to the motion of the wave itself
wave speed equation
v = λf
v = wave speed
λ = wavelength
f = frequency
transverse wave function moving in the +x direction
y(x, t) = A cos [ 2π (x/λ - t/T) ]
= Acos[kx - ωt)
A = amplitude
x = position
λ = wavelength
T = period
k = 2π /λ
transverse wave fcn moving in the - x direction
y(x,t) = Acos(kx +ωt)
transverse velocity
v_y(x,t) = ωAsin(kx - ωt)
- it’s the derivative of the transverse wave fcn
transverse acceleration
a_y(x,t) = -ω^2y(x,t)
if the tension is increase what happens to the wave speed?
the wave speed increases
transverse impulse is equal to…
transverse momentum
wave speed equation
v = sqrt (F/ μ)
F = tension in string
μ = mass per unit length
v = sqrt( restoring force returning system to equilibrium/ inertia resisting return to equilibrium)
power for sinusoidal waves
P(x,t) = sqrt(μF)ω^2A^2sin(kx - ωt)
μ = mass per unit length
F = tension in string
ω = wavelength
A = amplitude
k= force constant
x= displacement
t= time
max value for power occurs at
P_max = sqrt(μF)ω^2A^2
average power occurs at
P_av = (1/2)sqrt(μF)ω^2A^2
P_av = average power, sinusoidal wave on a string
μ = mass per unit length
F = tension in string
ω = angular frequency
A = wave amplitude
Inverse - square law for intensity
I_1/I_2 = r^2_2/r^2_1
intensity
the time average rate at which energy is transported by the wave per unit
intensity is inversely proportional to…
the squared distance from the source
principle of superposition
as two pulse overlap and pass each other, the total displacement of the string is the algebraic sum of the displacement at that point in the individual pulses
nodes
points that don’t move at all
antinodes
midway between nodes where amplitude is greatest
destructive interference
happens at nodes where two waves are always equal and opposite therefore they cancel out
constructive interference
happens at antinodes, the two waves are identical and result in a larger displacement
standing wave on a string, with a fixed end at x =0
y(x,t) = (A_sw*sinkw)sinwt
A_sw = standing wave amplitude = 2A
k = wave number
ω = angular frequency
adjacent nodes are one half wavelength apart.
whats the equation to find it
λ_n = 2L/n
whats the general equation for harmonics ?
f_n = n(v/(2L))
= nf_1
f_n = standing wave frequencies, string fixed at both ends
f_1 = fundamental fequency
n = 1,2,3,4…
v = wave speed
L = length of string
fundamental frequency equation
f_1 = 1/(2L)sqrt(F/μ)
= v/(2L)
audible sound range for humans
20-20000 Hz
ultrasonic vs. infrasonic
ultra = above audible range
infra = below audible range
P_max , pressure amplitude of sinusoidal wave fcn
P_max = BkA
B = bulk modulus
k = wave number = 2π/λ
