An explanation of standing waves on strings

1. Standing Waves on Strings
Jeremy Soroka

2. Consider this red string of length L attached with both ends fixed to two
wooden poles.
Let this point
of the string
be x=0
And let this
point of the
string be x=L
The two ends of the string will not move since they are fixed, therefore
the amplitude at x=0 and x=L is 0.
A x = 2Asin(
2𝜋
λ
L)=0  sin(
2𝜋
λ
L)=0 
2𝜋
λ
L = m𝜋,
where m is a positive, non-zero integer which makes sense
because sin(m𝜋) will always equal zero

3. Rearranging the equation, a string attached at two fixed points can oscillate in a standing
wave pattern only with certain wavelengths:
2𝜋
λ
L = m𝜋  λ=
2𝐿
𝑚
Q. Using the formula, what would the wavelengths be at m=0,
m=5, and m=10?

4. A.
m=0 isn’t an allowed value, so we can’t solve for that
m=5 is an allowed value so λ=
2𝐿
𝑚
=
2𝐿
5
m=10 is an allowed value so λ=
2𝐿
𝑚
=
2𝐿
10
Note that these standing waves are called normal modes
of vibration of the string
λ=
2𝐿
𝑚
Q. Using the formula, what would the wavelengths be at m=0,
m=5, and m=10?

5. 𝑓 =
𝑣
λ
=
𝑚
2𝐿
𝑣 =
𝑚
2𝐿
𝑇
μ
The frequencies associated with the normal modes of
vibration are:
Q. What wavelength corresponds to the lowest frequency?
A. Since frequency is inversely proportional to wavelength,
frequency will be lowest at the highest wavelength which would be
λ=2L
This frequency is called the fundamental frequency or the first
harmonic.

6. 𝑓 =
𝑣
λ
=
𝑚
2𝐿
𝑣 =
𝑚
2𝐿
𝑇
μ
Q. Using the formula, what could you do to the
following to increase the frequency?
a) Tension (T)
b) Length (L)
c) Linear mass density (μ)

7. 𝑓 =
𝑣
λ
=
𝑚
2𝐿
𝑣 =
𝑚
2𝐿
𝑇
μ
Q. Using the formula, what could you do to the
following to increase the frequency?
a) Tension (T)
b) Length (L)
c) Linear mass density (μ)
A.
a) Frequency is proportional to the square root of tension in the string, so it
could be increased by increasing the tension
b) Frequency is inversely proportional to length, so frequency increases when
the string is shorter
c) Frequency is inversely proportional to the square root of linear mass
density, so by choosing a string with a lower linear mass density, the
frequency will be higher

8. The other allowed frequencies, called harmonics or resonant
frequencies, can also be written in terms of the fundamental
frequency:
Eg. fm=mf1, where f1 is the fundamental frequency