Waves

A stretched wire has a fundamental frequency f.
When it is divided into 3 segments, their fundamental frequencies are f1,f2, and f3.
Tension T is constant.
We need to find the relation among f,f1,f2,f3.
Step 1: Formula for fundamental frequency of a stretched string
For a string under tension T, length L, and linear density µ :
f=‌

1

2L

√‌

T

µ

Step 2: Frequencies of the three parts
Suppose the original length is divided into three parts of lengths L1,L2,L3, so that
L=L1+L2+L3
Then their fundamental frequencies are
f1=‌

1

2L1

√‌

T

µ

,‌‌f2=‌

1

2L2

√‌

T

µ

,‌‌f3=‌

1

2L3

√‌

T

µ

Step 3: Express Li in terms of fi
From the above,
Li=‌

1

2fi

√‌

T

µ

and
L=‌

1

2f

√‌

T

µ

Step 4: Use L=L1+L2+L3
‌

1

2f

√‌

T

µ

=‌

1

2

√‌

T

µ

(‌

1

f1

+‌

1

f2

+‌

1

f3

)
Cancel the common factor ‌

1

2

√‌

T

µ

:
‌

1

f

=‌

1

f1

+‌

1

f2

+‌

1

f3