Standing waves — nodes, antinodes, harmonics for string, open pipe, closed pipe

Question

Compare the harmonic series for (a) a string fixed at both ends, (b) an open pipe, and (c) a closed pipe. Which harmonics are present in each case, and what are the fundamental frequencies?

(JEE Main & NEET — asked almost every year)

Solution — Step by Step

Fixed end / closed end: Must be a node (zero displacement)
Free end / open end: Must be an antinode (maximum displacement)

These constraints determine which standing wave patterns can fit in each system.

Both ends are nodes (string) or both ends are antinodes (open pipe). Either way, all harmonics are present.

f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \ldots

Fundamental: $f_1 = \dfrac{v}{2L}$

Closed end = node, open end = antinode. Only odd harmonics are present.

f_n = \frac{nv}{4L}, \quad n = 1, 3, 5, \ldots

Fundamental: $f_1 = \dfrac{v}{4L}$

The closed pipe’s fundamental is half that of an open pipe of the same length.

System	Boundary	Fundamental	Harmonics present
String (both fixed)	N — N	$v/(2L)$	All ( $1, 2, 3, \ldots$ )
Open pipe	A — A	$v/(2L)$	All ( $1, 2, 3, \ldots$ )
Closed pipe	N — A	$v/(4L)$	Odd only ( $1, 3, 5, \ldots$ )

Why This Works

Standing waves form when a wave reflects back and forth, creating a stable pattern of nodes and antinodes. The boundary conditions act as filters — only wavelengths that satisfy the boundary conditions survive.

graph TD
    A["Standing Wave Problem"] --> B{"Identify boundaries"}
    B -->|"Both fixed/both open"| C["Both N or both A"]
    B -->|"One fixed, one open"| D["One N, one A"]
    C --> E["All harmonics<br/>f = nv/2L"]
    D --> F["Odd harmonics only<br/>f = nv/4L"]
    E --> G["Nodes = n+1<br/>Antinodes = n"]
    F --> H["n = 1,3,5...<br/>Missing even harmonics"]
    A --> I{"Overtone or Harmonic?"}
    I --> J["pth overtone =<br/>(p+1)th harmonic<br/>(for all harmonics)"]
    I --> K["pth overtone =<br/>(2p-1)th harmonic<br/>(for odd only)"]

For the closed pipe, fitting one quarter-wavelength gives the fundamental. The next pattern that fits is three quarter-wavelengths (3rd harmonic), then five quarter-wavelengths (5th harmonic). Even harmonics don’t fit because they’d require a node at the open end or an antinode at the closed end.

Alternative Method — Drawing the Patterns

Sketch the wave shapes: draw nodes as dots and antinodes as loops.

String (n=1): N-loop-N (half wavelength fits)
String (n=2): N-loop-N-loop-N (full wavelength fits)
Closed pipe (n=1): N-loop (quarter wavelength fits)
Closed pipe (n=3): N-loop-N-loop (three-quarter wavelength fits)

Counting loops immediately tells you the harmonic number.

For NEET MCQs: if a closed pipe and an open pipe have the same fundamental frequency, then $v/(4L_c) = v/(2L_o)$ , giving $L_c = L_o/2$ . The closed pipe is half the length. This length comparison question appears frequently.

Common Mistake

The overtone-harmonic confusion strikes again here. For a closed pipe, the 1st overtone is the 3rd harmonic (not 2nd), the 2nd overtone is the 5th harmonic, etc. For a string or open pipe, the 1st overtone is the 2nd harmonic. Students mix these up constantly. Write out the sequence before answering: closed pipe harmonics are $1, 3, 5, 7, \ldots$ — the $p$ -th overtone is the $(2p+1)$ -th harmonic.

Question

Solution — Step by Step

Why This Works

Alternative Method — Drawing the Patterns

Common Mistake

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