Why does the frequency double when I halve the length of the string?

The fundamental frequency is inversely proportional to the length (f ∝ 1/L). Halving the length doubles the frequency, which corresponds to raising the pitch by one octave. This is because the wavelength of the fundamental standing wave is twice the string's length (λ = 2L), so a shorter string means a shorter wavelength and, for a wave with a fixed speed, a higher frequency.

Does this model explain why real guitar strings sound different even when playing the same note?

Partially. The fundamental frequency determines the note's pitch, but the timbre (sound quality) is shaped by the relative strengths of the harmonics. A real string's material, thickness (affecting μ), and flexibility influence which harmonics are emphasized and how they decay. This simulator focuses on the fundamental pitch and ideal harmonic patterns, simplifying timbre effects.

What is the difference between 'plucking' and showing the 'standing wave shape'?

Plucking simulates the initial excitation, which typically contains a mix of many harmonics. The resulting complex vibration is a superposition of these standing wave modes. The 'standing wave shapes' view isolates and displays the pure shape of individual harmonics (1st, 2nd, 3rd, etc.), helping you visualize the node and antinode patterns for each resonant frequency.

Why does increasing the tension raise the pitch?

Increasing tension increases the speed of waves traveling along the string (v = √(T/μ)). For a fixed length (and thus a fixed wavelength for the fundamental), the frequency is directly proportional to wave speed (f = v/λ). A higher wave speed therefore results in a higher frequency and a higher-pitched sound.

Monochord / Sonometer

f₁ = (1/2L)√(T/μ), harmonics, note label, Pluck; vs Standing Waves shapes.

Live graphs

About this model

A monochord, or sonometer, is a classic apparatus for demonstrating the fundamental relationship between the physical properties of a string and the pitch of the sound it produces. This simulator models a string fixed at both ends, which can be plucked to generate standing waves. The core physics is governed by the wave equation on a string, leading to the formula for the fundamental frequency: f₁ = (1/(2L))√(T/μ). Here, f₁ is the frequency, L is the vibrating length of the string, T is the tension, and μ is the linear mass density (mass per unit length). The model shows how changing any of these parameters—by adjusting sliders for tension, length, or density—directly alters the pitch, visually and audibly. The simulator simplifies real-world conditions by assuming a perfectly flexible, homogeneous string with ideal fixed boundaries that reflect waves without loss, and it neglects effects like stiffness and air damping. By interacting, students learn to predict how frequency scales with inverse length (f ∝ 1/L), with the square root of tension (f ∝ √T), and with the inverse square root of density (f ∝ 1/√μ). They can also explore harmonics by exciting the string at different modes, observing the corresponding standing wave patterns with nodes and antinodes, and connecting the harmonic series (f_n = n*f₁) to musical intervals. This provides a concrete foundation for understanding wave superposition, resonance, and the physics of musical instruments.

Who it's for: High school and introductory college physics students studying waves, sound, and harmonics, as well as music students learning the physical basis of pitch and string instruments.

Key terms

Standing Wave
Fundamental Frequency
Harmonic
Linear Mass Density
Tension
Node
Antinode
Sonometer

How it works

A monochord / sonometer idealizes a string between two bridges: wave speed c = √(T/μ) from tension and linear density, and fₙ = nc/(2L) for harmonics on a fixed–fixed string. Standing Waves in this lab emphasizes mode shapes; this simulator ties L, T, μ to pitch and optional Pluck tone.

Key equations

f₁ = (1/2L) √(T/μ), fₙ = n f₁, λₙ = 2L/n

Frequently asked questions

Why does the frequency double when I halve the length of the string?: The fundamental frequency is inversely proportional to the length (f ∝ 1/L). Halving the length doubles the frequency, which corresponds to raising the pitch by one octave. This is because the wavelength of the fundamental standing wave is twice the string's length (λ = 2L), so a shorter string means a shorter wavelength and, for a wave with a fixed speed, a higher frequency.
Does this model explain why real guitar strings sound different even when playing the same note?: Partially. The fundamental frequency determines the note's pitch, but the timbre (sound quality) is shaped by the relative strengths of the harmonics. A real string's material, thickness (affecting μ), and flexibility influence which harmonics are emphasized and how they decay. This simulator focuses on the fundamental pitch and ideal harmonic patterns, simplifying timbre effects.
What is the difference between 'plucking' and showing the 'standing wave shape'?: Plucking simulates the initial excitation, which typically contains a mix of many harmonics. The resulting complex vibration is a superposition of these standing wave modes. The 'standing wave shapes' view isolates and displays the pure shape of individual harmonics (1st, 2nd, 3rd, etc.), helping you visualize the node and antinode patterns for each resonant frequency.
Why does increasing the tension raise the pitch?: Increasing tension increases the speed of waves traveling along the string (v = √(T/μ)). For a fixed length (and thus a fixed wavelength for the fundamental), the frequency is directly proportional to wave speed (f = v/λ). A higher wave speed therefore results in a higher frequency and a higher-pitched sound.

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