Ch 15: Waves Flashcards
mechanical wave
disturbance that travels through a medium
transverse wave
displacements of medium are perpendicular to direction of travel of the wave
longitudinal wave
motions of the particles of the medium are back and forth along the same direction that the wave travels
wave speed
speed at which the wave propagates through the medium
T/F: Wave speed is not the same as the speed with which the individual particles move.
TRUE TRUE
T/F: The medium itself does travel through space when there is a wave going through it.
FALSE
Waves transport _____, not _____, from one region to another.
energy; matter
On a periodic transverse wave, each particle undergoes periodic motion with _____.
the same frequency, and wave speed is constant.
wave function for a sinusoidal wave moving in the +x-direction
y(x,t) =Acos(kx - ωt), where y is the displacement of a particle at time t and position x
wave function for a sinusoidal wave moving in the -x-direction
y(x,t) =Acos(kx + ωt)
compression
region of increased density
rarefaction
region of reduced density
wavelength of a periodic longitudinal wave
distance from compression to compression or rarefaction to rarefaction
wave number
k = 2π/λ
At time t, one point has maximum positive displacement while another point has maximum negative displacement- these two are a ______ cycle ______phase
half; out of
In the wave function, when y=A, what is the motion of the particle?
It is instantaneously at rest
wave speed for periodic transverse wave
v = λ/T = λf; generally determined by the mechanical properties of the medium and is constant (so increasing f causes λ to decrease and waves of all frequencies propagate with the same wave speed)
For a periodic wave, ω =
vk
If you use the wave equation to plot y as a function of x for time t, the curve shows _____.
the shape of the string at t
If you use the wave equation to plot y as a function of t for position x, the curve shows _____.
the displacement y of a particle at that coordinate as a function of time
phase
(kx ± ωt); plays the role of an angular quantity; determines what part of the sinusoidal cycle is occuring
What are the possible values of the phase for a crest (y = A)?
0, 2π, 4π, …
What are the possible values of the phase for a trough (y = -A)?
π, 3π, 5π, …
If you take the derivative of the phase, what do you get?
dx/dt = ω/k = v; aka phase velocity or speed
Given y(x,t) = Acos[2π/λ(x−vt)], what is an expression for the transverse velocity v_y at a fixed point x?
Take the partial derivative with respect to t:
v_y = (2πvA)/λ sin[2π/λ(x−vt)] or ωAsin (kx - ωt)
Given v_y = (2πvA)/λ sin[2π/λ(x−vt)], what is the maximum transverse speed of a particle on a string?
The speed is maximized when the sin term = 1, so v_max = (2πvA/λ), or ωA
What is the transverse acceleration (a_y) of a particle on a string?
Take the second partial derivative of y(x,t) with respect to t:
-ω²Acos(kx-ωt), which is simply -ω²y(x,t)
If you take the first partial derivative of the wave equation with respect to x, what do you get?
the slope of the string at point x and time t, -kA sin(kx-ωt)
If you take the second partial derivative of the wave equation with respect to x, what do you get?
the curvature of the string at point x and time t, -k²Acos(kx-ωt), which is simply -k²y(x,t)
What are the physical quantities that determine the speed of transverse waves on a string?
tension, mass per unit length (aka linear mass density)
Increasing the tension in a string increases the restoring forces that tend to straighten out the string when it is disturbed, thus increasing the _____.
wave speed
Increasing the mass of a string decreases the _____.
wave speed
The momentum of the moving portion of a string increases with time because _____.
more mass is brought into motion- the wave propagates at a constant speed v and so the increase in momentum is not due to an increasing v
What is the equation for the speed of a transverse wave on a string?
v = √(T/μ), where T is the tension and μ is the mass per unit length
What is the general form for wave speed?
v = √(restoring force returning the system to equilibrium/inertia resisting the return to equilibrium)
What is the power at a point on a sting?
the transverse force multiplied by the transverse velocity; the instantaneous rate at which energy is transferred along the string;
P(x,t)= F_y(x,t) v_y(x,t), where F_y is equal to the negative slope of the string (negative first partial derivative of the wave function with respect to x) at that point times the force, and v_y is equal to the first partial derivative of the wave function with respect to t
What is an alternate form of the power expression (using the wave function form)?
P(x,t) = √(μF) ω²A²sin²(kx-ωt)
What is the maximum instantaneous power, P_max?
P_max = √(μF) ω²A²
What is average power, P_av?
P_av = ½ √(μF) ω²A², since the average value of the sin² function is ½
intensity
average power per unit area, usually measured in watt per square meter; if the waves spread out equally in all directions, the intensity at distance r is:
I= P/(4πr²)
What is the inverse square law for intensity?
I_1/I_2 = r²_2/r²_1 if no energy is absorbed between the two spheres, making the power P the same for both
How do you find the frequency from the wave equation y(x,t) =Acos(kx - ωt)?
ω is equal to 2πf, so f =ω/2π
How do you find the wavelength from the wave equation y(x,t) =Acos(kx - ωt)?
k is equal to 2π/λ, so λ=2π/k
What are the units for μ?
kg/m
What are the units for T?
seconds/cycle
What are the units for f?
cycles/second
What are the units for ω?
rad/s
What are the units for intensity?
W/m²
How do you find the tension in a rope from the wave equation y(x,t) =Acos(kx - ωt)?
You find f by using ω=2πf, λ by using k=2π/λ, and v by using v=λf. Then you use the equation v = √(T/μ)
When there are two speakers out of phase facing each other, at what points will the sound be the loudest?
at the nodes
When a wave reflects from a fixed end, the pulse _____. Why?
inverts as it reflects; if a string exerts an upward force on a wall, the wall exerts a downward reaction force on the string, and so when the string is reflected, it travels on the opposite side of the string
When a wave reflects from a free end, the pulse _____. Why?
reflects without inverting; if a string reaches a free end, the free end exerts no transverse force on the string, and so the wave is reflected on the same side of the string
When a wave strikes the boundaries of its medium, all or part of the wave is _____.
reflected
interference
overlapping of waves
principle of linear superposition of waves
when 2 or more waves overlap, the total displacement is the sum of the displacements of the individual waves (add the two wave functions); y(x,t) = y_1(x,t) + y_2(x,t)
node
on a string, this is a point at which the string never moves
antinode
on a string, this is a point halfway between nodes and is where the amplitude of the string’s motion is the greatest
What is the distance between nodes or the distance between antinodes
½λ
What is the wave function for a standing wave on a string?
y(x,t) = (A_sw sin kx)sin ωt, where A_sw = 2A
Where are the nodes of a standing wave on a string, with fixed end at x=0?
0, λ/2, λ, 3λ/2, …
Unlike a traveling wave, a standing wave _____.
does not transfer energy from one end to the other
