A standing wave is a fussy guest. It cannot settle just anywhere — the ends of the string or pipe lay down the law. Tie an end down and the wave is forced to be still there. Let an end move freely and the wave insists on swinging hardest there. Those two demands are the boundary conditions, and they decide which standing waves are even allowed to exist.
📘 What you need to know
Standing waves form on strings and in pipes — a progressive wave superposes with its own reflection
A fixed end (string) or closed end (pipe) must be a node
A free end (string) or open end (pipe) must be an antinode
At a fixed end the reflected wave is inverted (anti-phase). At a free end it comes back upright (in phase)
Strings can be fixed–fixed, free–free, or fixed–free. Pipes can be closed–closed, open–open, or open–closed
Only certain frequencies fit the ends. These are the natural frequencies
A string’s frequency depends on its tension and its mass per unit length
Why an end forces the wave’s hand
Everything on this page comes from one question: what happens to the wave when it hits the end?
At a fixed end the string physically cannot move. The reflected wave comes back upside down, so incident and reflected always cancel there — a node
At a free end nothing holds the string back. The reflected wave comes back the same way up, so incident and reflected add there — an antinode
The ring on the rod is the classic picture of a free end: the string can still slide up and down there, so nothing forces it to be still.
Fixed / closed end cannot move
reflection inverted
Waves cancel destructive
so
NODE
Free / open end free to move
reflection upright
Waves add constructive
so
ANTINODE
Two words, and you never have to think again: tied means still. If an end is tied down, closed off, clamped, it is a node. If it is loose, open, free to wobble, it is an antinode. Every standing wave question begins by asking yourself this about both ends.
Boundary conditions on a string
A string has two ends, and each one can be fixed or free. That gives three combinations. Here is the simplest pattern each one allows.
Notice the bottom string: only a quarter of a loop fits. A fixed–free string is a different animal from a fixed–fixed one, and its allowed frequencies are different too.
The guitar connection: a guitar string is fixed at both ends, so both ends are nodes. How fast it vibrates depends on the tension (set by the tuning pegs) and the mass per unit length (which is why the low strings are the thick ones).
Boundary conditions in a pipe
Blow across the top of a bottle and the air column inside vibrates. These are longitudinal standing waves, but the rules are identical: closed end, node. Open end, antinode.
Careful: the wiggly line is a graph of air displacement along the pipe. The air itself sloshes back and forth along the tube, not up and down.
End of the medium
Reflected wave
What forms there
Fixed end of a string
Inverted — in anti-phase
Node
Closed end of a pipe
Inverted — in anti-phase
Node
Free end of a string
Upright — in phase
Antinode
Open end of a pipe
Upright — in phase
Antinode
Only certain waves are allowed
Now put the two ideas together. The ends demand nodes or antinodes in particular places, and nodes are spaced λ/2 apart. Between them, they leave only a short list of wavelengths that fit.
Any wavelength that fits the ends is allowed. Everything else dies away
The frequencies that go with those wavelengths are the natural frequencies of the string or pipe
Drive the system at one of them and you get a big, stable standing wave
🧭 Working out what fits
Label both ends first. Fixed or closed → N. Free or open → A.
Sketch the simplest curve that obeys both labels. Don’t add loops you don’t need.
Measure it in quarters. N to N is λ/2. N to A is λ/4. A to A is λ/2.
Set that equal to L and solve for λ. Only then use v = fλ.
WE 1
A pipe of length 0.85 m is open at one end and closed at the other. State the boundary condition at each end and determine the longest wavelength that can form a standing wave in the pipe.
Step 1 — label the ends
Open end → antinode. Closed end → node.
Step 2 — the simplest pattern that fits
Just a node at one end and an antinode at the other, with nothing in between.
That distance is a quarter of a wavelength: L = λ/4Step 3 — rearrangeλ = 4L = 4 × 0.85λ = 3.4 mLongest wavelength means simplest pattern. Add loops and the wavelength only gets shorter.
WE 2
A string of length 1.5 m is fixed at both ends. A standing wave on it has a total of 3 nodes. The speed of waves on the string is 60 m s⁻¹. Determine the wavelength and the frequency of the vibration.
Step 1 — turn nodes into loops
3 nodes (one at each end, one in the middle) → 2 loopsStep 2 — each loop is half a wavelengthL = 2 × λ/2 = λλ = 1.5 mStep 3 — use the wave equationf = v/λ = 60 / 1.5f = 40 HzBoth ends fixed, so both ends are nodes. That is what made “3 nodes” mean “2 loops”.
WE 3
A pipe of length 0.34 m is open at both ends. Taking the speed of sound in air as 343 m s⁻¹, calculate the lowest frequency at which a standing wave can form in the pipe.
Step 1 — label the ends
Both open → an antinode at each end, so one node sits in the middle.
Step 2 — antinode to antinode is half a wavelengthL = λ/2 → λ = 2 × 0.34 = 0.68 mStep 3 — use the wave equationf = v/λ = 343 / 0.68f = 504 HzLowest frequency goes with the longest wavelength, which is the simplest pattern. That is why we drew only one node.
💡 Top tips
Label the ends before you do anything else. N for fixed or closed, A for free or open.
Node to antinode is λ/4. Half the mistakes on this topic come from forgetting that.
Longest wavelength = simplest pattern = lowest frequency. Draw the fewest loops the ends allow.
A pipe diagram’s curve is a displacement graph, not the shape of the air.
An open–closed pipe is the odd one out: its simplest pattern is a quarter of a wavelength.
⚠ Common mistakes
Putting a node at an open end of a pipe — an open end is an antinode
Forgetting that a closed pipe end behaves exactly like a fixed string end
Thinking the reflected wave is inverted at every boundary — only at fixed and closed ends
Using L = λ/2 for an open–closed pipe. It is L = λ/4
Reading a pipe’s displacement curve as a transverse wave — sound is longitudinal
Assuming any frequency will produce a standing wave. Only the ones that fit the ends will
Quick recap: The ends decide everything. A fixed string end or closed pipe end reflects the wave inverted, so it must be a node. A free string end or open pipe end reflects it upright, so it must be an antinode. Match those demands against the λ/2 and λ/4 spacings and only a short list of wavelengths survives — the natural frequencies of the string or pipe.
You have just found the simplest pattern for each set of ends. But the ends will happily accept more loops: two, three, ten. Each one is a new allowed wavelength and a new allowed frequency, and they come in a beautiful sequence. Those are the harmonics, and they are the next page.
Boundary conditions boxing you in?
Book a free meeting and we’ll practise labelling ends, sketching patterns and turning them into wavelengths and frequencies.
