Physics of Waves

1) A 1.10 m long glass tube is open at one end and closed at the other end by a piston whose position can be adjusted. A tuning fork emits a sound wave into the tube through its open end with a frequency 694 Hz.

A) What is the maximum length of the tube that will allow resonance to occur?

B) What is the second length of tube that will allow resonance to occur?

 

2) $y(x,t)=(5.1 mm)*sin (kx+(730 rad/s)t+Phi)$ describes a wave traveling along a string. How much time is requires for any given point on the string to move from y=-2.0 mm to y=+2.0 mm?

1) the speed of sound in air at 20 degree C is $V =343 m/s$

the wavelength for a frequency. $F =694$ is $lambda = V*T=V/F (=343/694) =0.494 m$

In the tube open at one end and closed at the other end there will be $lambda/4$ for the fundamental (see figure at bottom) and $lambda/4 +k*lambda/2$ for other resonances

$L =lambda/4 = 0.1235 m$ for the fundamental

$lambda/2 =0.494/2 =0.247 m$

we have the condition

$lambda/4 +k*lambda/2 < 1.10$

$0.1235+k*0.247 <1.10$

$k < 3.95$

the longest length of tube for resonance is for $k =3$ (the 3rd resonance)

$L =0.1235+3*0.247 =0.8645 m$

the next resonance happens for $k =2$

$L =0.1235 +2*0.247 =0.6175 m$

2) $y(x, t) = (5.1 mm) sin(k x + (730 rad/s)t + phi)$

$y(x,t) =5.1*sin(k*x+730*t +phi)$

at the same x, the y varies from -2 mm to 2 mm.

we suppose $x =0$, $phi =0$

$y(t_1) =5.1*sin(730*t_1) =-2 mm 730*t_1 =-0.403 rad$,

$t_1 =-5.52*10^{-4} sec$

$y(t_2)=5.1*sin(730*t_2)=+2 mm 730*t_2 =+0.403 rad$

$t_2 =+5.52*10^{-4} sec$

$Delta(t) =2*5.52*10^{-4} =0.0011 sec =1.1 ms$