3.3 Waves - Standing Waves

Flashcards covering standing waves, nodes/antinodes, harmonics for open/open and open/closed pipes, wave speed determinants, beats, and timbre.

What are standing waves?

Stationary waves formed when two waves with the same speed and frequency travel in opposite directions and interfere, resulting in a pattern that does not travel along the medium; the displacement is the sum of the two waves.

How does interference differ for waves moving in the same direction versus opposite directions?

Same direction interference yields a travelling wave; opposite directions interference yields an oscillating wave fixed in space (a standing wave).

What is a node in a standing wave?

A point that remains at rest (zero displacement) at all times.

What is an antinode in a standing wave?

A point with maximum displacement in the standing wave.

What is the distance between two adjacent nodes in a standing wave?

λ/2.

How is the wavelength related to the distance between adjacent nodes in a standing wave?

λ = 2 × (distance between adjacent nodes).

For a standing wave on a string fixed at both ends, what boundary condition exists at the ends?

Nodes at both ends.

In an open-open pipe, which harmonics are possible?

All harmonics are possible; L = nλ/2 and f_n = n v / 2L.

In an open-open pipe, what is the fundamental wavelength relative to the length L?

λ1 = 2L (fundamental wavelength).

In an open-open pipe, what is the fundamental frequency?

f1 = v / 2L (and f_n = n v / 2L for all harmonics).

In an open/closed pipe, what are the boundary conditions?

Node at the closed end and antinode at the open end.

Which harmonics are allowed in an open/closed pipe?

Only odd harmonics (1st, 3rd, 5th, …).

In an open/closed pipe, what is the fundamental wavelength?

λ = 4L (L = λ/4).

In an open/closed pipe, what is the fundamental frequency?

f1 = v / 4L; higher odd harmonics: f_n = n v / 4L with n odd.

What determines wave speed on a string?

Tension and linear density μ; v = sqrt(T/μ).

How does increasing tension affect wave speed and frequency when wavelength is fixed?

Increases wave speed; since v = fλ, frequency increases if λ is fixed.

How does increasing density affect wave speed and frequency?

Increases density (μ) -> decreases v; with fixed λ, frequency decreases.

What is timbre?

The characteristic quality of a sound that allows us to distinguish between different instruments playing the same pitch, due to differences in harmonic content (number and amplitude of harmonics).

Why do two instruments playing the same note sound different?

Because of differences in harmonic content and amplitudes, which produce different timbres.

What are beats?

Regular variations in loudness that occur when two waves have slightly different frequencies; beat frequency is |f1 − f2|.

How are beats used to tune instruments?

Two sounds are played together until the beats disappear, indicating the frequencies match.

How does the superposition principle relate to standing waves?

The resultant wave is the sum of the individual waves; standing waves arise from the interference of two waves of equal speed and frequency traveling in opposite directions.

In an open-open standing wave, what do the ends represent?

The ends are antinodes (points of maximum displacement); the ends are open to move air freely.

In an open/closed pipe, what does a node at the closed end imply?

Displacement is zero at the closed end (node).

Why are only odd harmonics possible in open/closed pipes?

Because a closed end must be a node and an open end an antinode; only odd-harmonic patterns satisfy this boundary condition (quarter-wave segments).

What is the relation between length, pitch, and frequency in strings/pipes?

Longer length lowers frequency (lower pitch); shorter length raises frequency.

What is a half-wavelength segment in a standing wave on a string?

The segment between two adjacent nodes is half a wavelength long; particles inside this segment are in phase, while adjacent segments are out of phase.