(i) A person hums in a wall and finds strong resonance at frequencies `60 Hz, 100Hz` and `140 Hz`. What is the fundamental frequency of the well? Explain? How deep is the wall ? (velocity of sound `= 344 m//`).
(ii) A violin string `33cm` long vibrates at a fundamental frequency of `440 Hz`. Where shold a finger be pressed on the string board so that its decreased length causes it to vibrate in its fundamental mode at five fourth of its original frequnency.

Answer

(i) fundamental frequercy `n = 20 Hz`, depth of the weil `= 4.3`, (ii) at `26.4 cm` from the end.

Explanation

(i) The well can be considered as an organ pipe closed at one end and open at the other. At the fundamental mode, `lambda//4 = l` (depth of the well in this case). The given frequencies happen to be `3rd, 5th` & `7th` harmonics of the fundamental frequency `n = 20 Hz`. Now use `v = nlambda` so as to obtain depth of the well `= 4.3 m`.
(ii) Use the law of vibrating string ni `=` constant. To have the frequency `5//4` th of original, the length requireed `4//5` of `33cm`. Thus, the finger must be pressed at a point will divid the length of the string in a ratio `1:4` is at `26.4 cm` from the end.