Standing waves on strings and harmonics

The boundary condition

• A string fixed at both ends cannot move at either end, so both ends must be nodes.
• Only the wave patterns that fit a whole number of half wavelengths between those two nodes can exist.
• Each allowed pattern is a harmonic (or mode), and each has its own natural frequency.

The harmonic series for a string

For a string of length $L$ with waves travelling at speed $v$:

$$\lambda_n = \frac{2L}{n} \qquad f_n = \frac{nv}{2L} = n f_1$$

• $n$ — the harmonic number ($n = 1, 2, 3, \dots$)
• $\lambda_n$ — the wavelength of the $n$th harmonic
• $f_n$ — the frequency of the $n$th harmonic
• $f_1$ — the fundamental frequency, or first harmonic

Harmonic	Loops	Wavelength	Frequency
1st (fundamental)	1	$2L$	$f_1 = \dfrac{v}{2L}$
2nd	2	$L$	$2f_1$
3rd	3	$\dfrac{2L}{3}$	$3f_1$
$n$th	$n$	$\dfrac{2L}{n}$	$nf_1$

• A string produces all harmonics — every whole-number multiple of the fundamental, because every whole number of half wavelengths fits between two nodes.
• Harmonic $n$ has exactly $n$ loops and $n + 1$ nodes (counting the two at the ends).

Deriving a harmonic rather than recalling it

The formulas above follow from a sketch in three steps, and this is the method to use in an exam:

• Draw the string with a node at each end.

• Fit the required number of loops between them, remembering each loop is $\lambda/2$.

• Solve for $\lambda$: if $n$ loops fit in length $L$, then $n\dfrac{\lambda}{2} = L$, so $\lambda = \dfrac{2L}{n}$.

• Convert to frequency with $v = f\lambda$.

• This works for any system, including ones the formula table does not cover, and it never leaves you guessing which formula applies.

What sets the wave speed

• The frequencies depend on the wave speed on the string, $v$, which is a property of the string itself — its tension and its mass per unit length.
• Increasing the tension increases $v$, so all the harmonic frequencies rise. This is how a guitar is tuned.
• A thicker, heavier string has a lower $v$, so its frequencies are lower — which is why the low strings on a guitar are the thick ones.
• Shortening the string (pressing a fret) reduces $L$, which raises $f_1 = v/2L$.
• At Level 3 you are not asked to calculate $v$ from tension and mass per unit length — that relationship is not on the resource sheet. You will always be given $v$, or be able to find it from $v = f\lambda$.

Why instruments sound different

• A plucked string vibrates in several harmonics at once, not just the fundamental.
• The pitch you hear is set by the fundamental; the timbre — what makes a violin sound unlike a guitar playing the same note — is set by the relative strengths of the higher harmonics.
• Where the string is plucked affects which harmonics are excited: plucking at the midpoint favours odd harmonics, because the midpoint is a node for every even harmonic and so cannot excite them.