Harmonics Flashcards
define beat
when you play two different waves that are VERY CLOSE in wavelength/frequency but not quite the same, creating a summed wave that passes through areas of constructive and destructive interference
what is simple harmonic motion?
anything with a LINEAR restoring force directly proportional to the displacement, centered around equilibrium
works with ANYTHING that has a restoring force, like homeostasis, blood pressure, or financial markets, or trying to push something back to equilibrium
if the force is roughly linear, you get this motion
Hooke’s law
F = -kx, negative because restoring force = always opposite to displacement
energy of a spring (U(s))
1/2 * k * x^2
x = distance from EQUILIBRIUM, thus AMPLITUDE
where is potential energy greatest at a spring pulled in fully at -a and spread out max at a? where is it going the fastest?
both, push vs. pull
fastest at midpoint between -a and a, or 0
where is force, acceleration, and velocity greatest, least, and 0 on a spring?
force and acceleration: greatest at ends, 0 at point 0 of equilibrium
velocity: greatest at 0, none at ends
CALCULUS TIME, derive F = -kx into a position function
F = -kx
ma = -kx
a = -k/mx
second deriv (x) w/ respect to time = -k/mx(t)
the original function must be sin or cos, making it negative
position as a function of time
x = Asin(omega * t)
or
x = A sin (2*pi/torque * time)
Period vs. frequency
inverses, period = seconds / cycle
frequency = cycles / second
period, T = 1/f
f = 1/T
Period of a spring vs. period of a pendulum
Tp = 2 * pi (sqrt(l / g))
Ts = 2 * pi * (sqrt(m/K))
k, spring constant
Newtons per meter
a 200 g mass is attached to a spring, creating a simple harmonic motion w/ a period of .25 s. if the total energy of the system = 2 J, find
(a) the force constant of the spring
(b) the amplitude of the motion
(a)
Ts = 2*pi * sqrt(m/k)
plug in and get 126.33
(b)
we know that at the turning point all of the energy is stored as potential energy, and thus the 2 J should be equal to the potential energy equation of the spring
2 = 1/2 * kx^2
and we know that AMPLITUDE is equal to the x as x = distance from equilibrium, so plug in k from part a and solve!
determine height
9.4 second period,
pendulum from a tower
how does tension impact wave speed?
increased tension = increased speed, thicker/denser materials = slower
speed = square root of (tension/mass/length)
AKA
sqrt tension/ linear mass-
longitudinal wave and the wavelength of it
AKA compression wave, fixed point (like coil on slinky) moving PARALLEL back and forth IN THE DIRECTION OF the wave motion, instead of peaks and valleys you have compressed areas and rarefactions (low density area)
wavelength = distance between compressions OR distance between rarefactions
EX. soound
compression vs. rarefaction
high molecular density/pressure vs. low molecular density/pressure
transverse wave
medium vibrates PERPENDICULAR to the wave direction (up and down light wave)
when thinking waves/discerning between types always think about how _______ is moving
a fixed point in the medium
interference pattern
waves from different sources arrive at the same place and time
constructive vs. destructive interference
constructive: two waves AT THE SAME LOCATION overlap (ex. peak + peak) and add to create a larger amplitude crest, amplitude = directly sum of waves A and B
destructive: two waves at OPPOSITE locations (ex. Peak + trough) collide to cancel each other out
when waves collide, do they bounce off?
no, they pass THROUGH each other as if they never collided (even in destructive interference, they pass through after they are no longer in the same place)
active vs. passive noice cancelling
active = creates opposite sounds to outside ambient noise
passive = insulation
what type of interference is “in phase?” which is “out of phase?’
constructive = in phase, everything lines up
destructive = out of phase, everything lines up to CANCEL OUT
when waves hit a wall/fixed source….
and when they have a “free end…”
they are inverted
they are not inverted
standing waves
special frequency of wave interference (constructive AND destructive) where there are nodes of complete destructive interference (nodes) and antinodes (complete CONSTRUCTIVE interference) where amplitudes are at minimums and maximums respectively
simplest standing wave and the frequency necessary to create it
half of a wavelength, just one peak
L = lambda/2
lambda = 2L
we know v = lambda * f
thus,
f = v/2L
standing wave equation
for the nth harmonic, (amount of wavelengths)
f = n * v / 2L
n = number of antinodes
beat frequency
at what frequency the final beat wave (summed both waves) it gets louder and quieter
how many times with a wave with a frequency of 303 Hz match a wave with 300 Hz?
3 times PER SECOND, think the runner analogy, its just the difference between the two wave/second values (f1 - f2)
Doppler effect
car eeeeeeeuuuuuuuuiyour sound has a higher frequency from the front than from behind because its moving, the sound waves are more compressed
a moving source emits waves that are close together in the direction it is moving (shorter wavelength, thus higher frequency)
how does the Doppler effect work if sound has a constant speed?
because its the SOURCE catching up to the sound a little bit, after all sounds are emitted they move at the same speeed. this does not violated sound’s constant speed in air at all.
when does the Doppler effect take effect?
when EITHER/OR the source AND the hearer are moving, if you run towards the sound its still higher freq.
explain the sonic boom
if the source is moving at the speed of sound or faster, it causes an EXTREME form of the Doppler effect. as you keep moving, the peaks keep piling up on top of each other, creating a MASSIVE soundwave
equation for Doppler effect frequency
f= frequency of what observer will hear
f0 = frequency of source
v = speed of wave through medium
v0 = speed of observer
vs = speed of source
f = f0( (v +/- v0) / (v -/+ vs)
- = source going toward observer
+ = source going away from observer
vs OR v0 = 0 if the source or observer is not moving
use the Doppler effect:
what frequency does an observer hear if an 800 Hz whistle is:
(a) moving toward the listener at 10 m/s
(b) listener moving towards whistle at 10 m/s
(c) moving away from listener at 10 m/s
speed of sound = 343 m/s —> 340 m/s
a. f = 800 * (340/ (340-10))
f = 824 Hz
b. 824 Hz
c. 777 Hz
equations for frequencies in a string (held in place on both ends), air column (open/open) and another air column (open/closed)
string (held in place on both ends)
boundary conditions: nodes at both ends
fn = n * v/2l, n = 1, 2, 3….
air column (open/open)
boundary conditions: antinodes at both ends
fn = n * v/2L, n = 1, 2, 3….
air column (open/closed):
boundary conditons: node at closed/antinode at open
fn = n * v/4L, n = 1, 3, 5, 7….
