mechanical waves

11 flashcards · Physics · by inquisitive learner

instantaneous power in a wave
P(x,t)P(x,t)P(x,t) === FxF_xFx kkk ω\omegaω A2A^2A2 sin⁡2(kx−ωt)\sin^2{(kx - \omega t)}sin2(kx−ωt) == = μF\sqrt{\mu F}μF ω2\omega ^2ω2 A2A^2A2 sin⁡2kx−ωt\sin^2{kx - \omega t}sin2kx−ωt
maximum power
P(x,t)P(x,t)P(x,t) === μF\sqrt{\mu F} μF ω2\omega^2ω2 A2A^2A2
average power of a wave
⟨P⟩\lang P \rang⟨P⟩ === 12\frac{1}{2}21 μF\sqrt{\mu F}μF ω2\omega^2 ω2 A2A^2A2 ⟨P⟩\lang P \rang⟨P⟩ === average power 12=\frac{1}{2}=21= ⟨sin⁡2kx−ωt\lang \sin^2{kx - \omega t}⟨sin2kx−ωt ⟩\rang⟩ μ=\mu = μ= mass per unit length F=F =F= force ω\omegaω === angular velocity A=A =A= wave amplitude
impedance definition
force/velocity; the amount of force required to impart a certain velocity
transverse velocity
speed of material at a single location
amplitude transmission coefficient
t=t =t= 2Z1Z1+Z2\frac{2Z_1}{Z_1 + Z_2}Z1+Z22Z1 t=t =t= amplitude transmission coefficient Z1=Z_1 =Z1= impedance of the first material Z2=Z_2 =Z2= impedance of the second material
Adjacent nodes are separated by half wavelength intervals
wave impedance equation (transverse wave_
ZZ Z === Fv\frac{F}{v}vF =μF=\sqrt{\mu F}=μF Z=Z=Z= impedance; F=F=F= force (transverse to the direction of propogation) v=v=v= velocity μ=\mu =μ= mass per unit length
amplitude reflection coefficient
r=r=r= Z1−Z2Z1+Z2\frac{Z_1 - Z_2}{Z_1 + Z_2}Z1+Z2Z1−Z2 r=r =r= amplitude reflection coefficient Z1=Z_1 =Z1= impedance of first material Z2=Z_2=Z2= impedance of second material
At nodes, sin (kx) = 0 kx = 0,π,2π,3π...0, \pi, 2\pi, 3\pi ...0,π,2π,3π... (integer multiples of pi) x=x =x= 0,λ2,λ,3λ20, \frac{\lambda}{2}, \lambda, \frac{3\lambda}{2}0,2λ,λ,23λ ......... {half-integer multiples of the wavelength)
mathematically
standing wave frequencies, string fixed at both ends:
fn=f_n =fn= nv2L\frac{nv}{2L}2Lnv =nf1= n f_1=nf1 f=f =f= frequency n=n=n= nth harmonic, (n-1) overtone v=v=v= wave speed L=L=L= length of string n2L\frac{n}{2L} 2Ln === λn\lambda_nλn