〜 Mechanical Waves

Wave on a string: A harmonic wave propagates on a taut string with speed $v=80$ m/s and frequency $f=200$ Hz. The string is $2$ m long with tension $F_T=64$ N. Find: (a) wavelength $\lambda$; (b) wave number $k$; (c) angular frequency $\omega$; (d) linear mass density $\mu$ of the string.

String resonance: A violin string of length $L=1.2$ m is under tension $F_T=48$ N. Its linear mass density is $\mu=3\times10^{-3}$ kg/m. Find: (a) the wave speed on the string; (b) the first three resonant frequencies; (c) the wavelength of the third harmonic.

Doppler — train and moving observer: A train emits a whistle at $f_0=520$ Hz and moves toward the observer at $v_s=10$ m/s. Simultaneously, the observer cycles away from the train at $v_o=2$ m/s. The speed of sound is $v=340$ m/s. Find the perceived frequency. What if the observer instead approached the train?

Decibels of a source: A point sound source emits isotropically with power $P=100$ W. Find: (a) the sound intensity $I$ at $r=50$ m; (b) the sound level $\beta$ in dB; (c) the distance at which the level drops to 60 dB (normal conversation).

Sound pipes: An organ pipe of length $L=0.85$ m is open at one end and closed at the other (open-closed pipe). The speed of sound is $v=343$ m/s. Find: (a) the first three resonant frequencies; (b) the fundamental frequency if the same pipe were open at both ends (open-open pipe). Explain why the timbre changes.