Consecutive standing waves occur at frequencies 280 Hz and 315 Hz on a taut

string fixed at both ends. Determine the frequency of the 4th harmonic.

Question

Consecutive standing waves occur at frequencies 280 Hz and 315 Hz on a taut

string fixed at both ends. Determine the frequency of the 4th harmonic.

Vocaloid · Answer

Nth harmonic frequency formula for a string w/ fixed ends:

fn = (nv)/(2L)

We have two consecutive frequencies but we don’t know what n-value either of them are. So we let the lower frequency 280 hz correspond to n and the higher frequency correspond to n + 1 giving us

280 = nv(2L)
315 = (n+1)v/(2L)

Divide the two equations to eliminate L and v. Solve for n.

Once you have your n value, you know that

n * (f1) = 280
Solve for the fundamental frequency f1. Finally, multiply 4 * f1 to find the fourth harmonic.

Vocaloid · Answer

correction: 280 = nv/(2L)
with division between the nv and the 2L

GoodFellow · Answer

Good Job Vocaloid. I knew the answer. 
For your understanding, I would say, (this is my way) just expand the second equation. Then you get the term corresponding to 280 in it. Just go ahead and subtract & get the value of$$\frac{v}{2L}$$. To get the final answer just multiply that by 4. 
Do not take effort to find n. 
Good Job! again.