Standing Waves on a String: Harmonics and Nodes | Physics Demo

Standing waves form in air columns with specific boundary conditions. For open-open pipes (open at both ends), fundamental wavelength is λ₁ = 2L, frequency f₁ = v/(2L), and harmonics are fₙ = n × v/(2L) for n = 1, 2, 3... For open-closed pipes (open at one end, closed at other), fundamental wavelength is λ₁ = 4L, frequency f₁ = v/(4L), and only odd harmonics are present: fₙ = n × v/(4L) for n = 1, 3, 5... The boundary conditions determine which harmonics are allowed.

In open-open pipes, both ends are antinodes, with fundamental wavelength λ = 2L and harmonics at 1, 2, 3, 4... In closed-open pipes, the closed end is a node and open end is an antinode, with fundamental wavelength λ = 4L and only odd harmonics (1, 3, 5, 7...). The formulas differ from strings due to different boundary conditions.

Standing waves form in air columns with different boundary conditions. In open-ended columns (open at both ends), antinodes form at both ends, allowing all harmonics. The wavelength is λ_n = 2L/n and frequency f_n = nv/2L. In closed-ended columns (closed at one end), a node forms at the closed end and antinode at the open end, allowing only odd harmonics. The wavelength is λ_n = 4L/n and frequency f_n = nv/4L for odd n.

Standing waves also form in air columns as longitudinal waves. Open-open pipes (open at both ends) produce all harmonics (n=1,2,3...) with fₙ = nv/(2L) and λₙ = 2L/n. Open-closed pipes (open at one end, closed at the other) produce only odd harmonics (n=1,3,5...) with fₙ = nv/(4L) and λₙ = 4L/n. The boundary conditions determine which harmonics are possible. For a string with 4 nodes (third harmonic), wavelength is λ₃ = 2L/3. These principles explain sound production in musical instruments like flutes and organ pipes.

For pipes with one end closed and one end open (like clarinets), the closed end must have a node (zero displacement) because air molecules cannot move at the closed end. The open end must have an anti-node. These different boundary conditions result in different standing wave patterns compared to open-ended pipes.

Stringed instruments (guitar, violin) have natural frequency depending on length, thickness, and tension. Formula: f = (1/2L)√(T/(πr²d)), where d is density. Plucking at different positions produces harmonics: middle (1/2L) gives fundamental (1f), 1/4L gives second harmonic (2f), 1/6L gives third harmonic (3f). Frequency ratios are 1:2:3, wavelength ratios are 3:2:1. Piano keys connect to different strings with varying tension, length, and cross-section.

The solution proves string instruments produce harmonic sounds: frequencies are integer multiples of the fundamental ω₁ = aπ/L. Musicians adjust tension T via tuning pegs to change pitch—higher tension increases frequency. Amplitudes depend on plucking position: maximum near ends, minimum at center. Rapid coefficient decay (1/n²) means low-order harmonics dominate sound. Old strings lose tension, causing flatness. This explains why string instruments generate pleasant, predictable tones.

There are two naming systems for standing wave frequencies: the fundamental mode (n=1) is the lowest frequency, with higher frequencies called first overtone (n=2), second overtone (n=3), etc. Alternatively, the fundamental is the first harmonic, the next is the second harmonic, and so on. The fundamental corresponds to one half-wavelength, the first overtone/second harmonic to one full wavelength, the third harmonic to one and a half wavelengths, and the fourth harmonic to two wavelengths. The resonant frequencies for strings are given by fₙ = nv/(2l), where v = √(T/μ) is the wave speed, T is tension, and μ is linear mass density. Three factors affect pitch: (1) String length has an inverse relationship with frequency—longer strings produce lower pitches (pianos have strings of different lengths, guitars use fingering to change effective length). (2) String tension has a direct relationship with frequency—higher tension produces higher pitches (instruments are tuned by changing tension). (3) String density has an inverse relationship with frequency—thicker/denser strings produce lower pitches (pianos and guitars have strings of varying thicknesses).

A string fixed at both ends vibrates in multiple modes called harmonics. The first harmonic (fundamental) has one antinode with λ₁/2 = L. The nth harmonic has n antinodes with nλₙ/2 = L, or λₙ = 2L/n. Frequency is fₙ = nv/(2L), directly proportional to harmonic number. Wave velocity is v = √(T/μ), where T is tension and μ is linear mass density. Three factors affect frequency: (1) Tension - higher tension increases frequency, (2) Linear mass density - thinner strings have higher frequency, (3) Length - shorter strings have higher frequency. This explains guitar tuning, string thickness differences between bass and treble strings, and how fretting changes pitch.

Guitar strings produce sound through superposition of harmonics. The fundamental (1st harmonic) has the lowest frequency with one antinode. Higher harmonics have frequencies that are integer multiples of the fundamental. The frequency ratio between adjacent semitones is 2^(1/12) ≈ 1.0595. To tune a string to 440 Hz when currently at 466 Hz (one semitone sharp), tension must be decreased since f ∝ √(F). The tension adjustment follows from the frequency ratio.

The particle in a box model confines a particle to a region with infinite potential barriers at x=±L/2. Inside, V=0; outside, V=∞. Wave function must vanish at boundaries. General solution: ψ(x) = C cos(kx) + D sin(kx). Boundary conditions ψ(0)=0 and ψ(L)=0 require C=0 and sin(kL)=0, giving k = nπ/L. Allowed energies: E_n = (n²π²ℏ²)/(2mL²). Wave functions: ψ_n(x) = √(2/L) sin(nπx/L). This demonstrates how boundary conditions quantize energy levels, analogous to standing waves on a string.

The particle in a box model confines a particle to a region of length L with infinite potential barriers. Inside the box (V=0), the Schrödinger equation gives ψ = Acos(kx) + Bsin(kx). Boundary conditions ψ(0)=0 and ψ(L)=0 require kL = nπ, quantizing k to k_n = nπ/L and energy to E_n = (n²π²ℏ²)/(2mL²). Normalized wavefunctions are ψ_n(x) = √(2/L)sin(nπx/L) for n = 1, 2, 3, ... Negative n values don't produce new physical states since they give the same wavefunction with opposite phase. This demonstrates how boundary conditions quantize energy levels, unlike classical particles which can have any energy.

The particle in a box model confines an electron between two infinite potential walls separated by distance D. The electron forms standing waves with wavelengths satisfying λ/2 × n = D, where n is the quantum number. This model demonstrates how boundary conditions quantize energy levels.

From k = nπ/a and E = ℏ²k²/(2m), the energies are Eₙ = n²π²ℏ²/(2ma²). These discrete energy levels arise naturally from boundary conditions. The ground state (n=1) has lowest energy, and higher states have energies proportional to n². This quantization is fundamentally quantum mechanical and wouldn't occur in classical physics.

When a particle is confined in a one-dimensional box, it forms standing waves. The standing wave pattern depends on the quantum number n, representing the number of loops or antinodes. For n=1, there is one loop with nodes at boundaries. For n=2, two loops with an additional node in the middle. For n=3, three loops with two additional nodes. The energy is quantized and given by E_n = (n²h²)/(8mL²), directly proportional to n². This differs from the Bohr atomic model where energy is proportional to 1/n². The energy is always positive and increases with the quantum number.