Question

Two strings on a musical instrument are tuned to play at 392 Hz(G) and 494 Hz (B).

Part A

What are the frequencies of the first two overtones for G?

Got this, the answer was 784 and 1180

Part B

What are the frequencies of the first two overtones for B?

Need this

Part C

If the two strings have the same length and are under the same tension, what must be the ratio of their masses mGmB?

Got this. It was 1.59

Part D

If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths ℓG/ ℓB?

Need this

Part E

If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?

Need this

I need part B, D, and E

Answer 1

part B:

fundamental frequency=494 Hz

first overtone=second harmonics=494*2=988 Hz

second overtone=third harmonics=494*3=1482 Hz

part D:

fundamental frequency of standing wave on a string is given by

f=speed of wave on the string/(2*length of the string)

speed of wave of the string=sqrt(tension/mass per unit length)

given that strings have same tension and mass per unit length,

speed of the wave will also be same

then as frequency is inversely proportional to the length of the string

f1/f2=L2/L1

where f1 and L1 are frequency and length of for the first string

and f2 and L2 are frequency and length of second string

given that f1=392 Hz, f2=494 Hz

then L2/L1=392/494

==>L2/L1=0.79352

we are asked to find LG/LB=L1/L2=1/0.79352=1.2602

part E:

as their lengths are the same and as fundamental frequency=speed of the wave /(2*length of the string)

fundamental frequency is directly proportional to the speed.

now, as speed=sqrt(tension/mass per unit length)

and the strings have same mass and same length,

mass per unit length is same for both the strings

then speed is directly proportional to the square root of tension

hence fundamental frequencies are also directly proportional to square root of tension

if tension in first string be T1 and tension in second string be T2

then 392/494=sqrt(T1/T2)

==>ratio of tensions=(392/494)^2=0.62968