Find F 1 Calculator

Calculate f1 (Fundamental Frequency)

Enter the properties of the string to calculate its fundamental frequency (f1) and related values.

Harmonic (n)	Frequency (fn)	Wavelength (λn)	Formula
1 (Fundamental)	—	—	f1 = v / (2L)
2 (1st Overtone)	—	—	f2 = 2 * f1
3 (2nd Overtone)	—	—	f3 = 3 * f1
4 (3rd Overtone)	—	—	f4 = 4 * f1

Table of the first four harmonics and their properties.

What is the Fundamental Frequency (f1)?

The fundamental frequency, often denoted as f1, is the lowest frequency at which a system (like a string, air column, or membrane) vibrates naturally when disturbed. It's the first harmonic of the system and represents the simplest mode of vibration.

When an object vibrates, it typically does so at a mixture of frequencies. The fundamental frequency (f1) is the dominant and lowest frequency in this mix, and it usually determines the perceived pitch of the sound produced by the vibrating object. Other frequencies present are called overtones or higher harmonics, and they are integer multiples of the fundamental frequency in ideal systems.

This Fundamental Frequency (f1) Calculator is specifically designed to find the fundamental frequency of a stretched string, which is relevant in musical instruments like guitars, pianos, and violins, as well as in physics demonstrations.

Who should use the Fundamental Frequency (f1) Calculator?

• Physics students studying waves and sound.
• Music students and musicians interested in the physics of their instruments.
• Engineers working with vibrating structures.
• Anyone curious about the relationship between physical properties and sound.

Common Misconceptions

One common misconception is that an object vibrates ONLY at its fundamental frequency. In reality, most vibrating objects produce a complex sound containing the fundamental and several overtones, which give the sound its characteristic timbre or quality.

Fundamental Frequency (f1) Formula and Mathematical Explanation

For a string fixed at both ends, like a guitar or piano string, the fundamental frequency (f1) is determined by its length (L), the tension (T) it's under, and its linear mass density (μ – mass per unit length).

The speed of a wave (v) traveling along the string is given by:

v = sqrt(T / μ)

The fundamental mode of vibration for a string fixed at both ends corresponds to a standing wave with a wavelength (λ1) that is twice the length of the string:

λ1 = 2L

The relationship between frequency (f), wavelength (λ), and wave speed (v) is:

v = f * λ

So, for the fundamental frequency (f1):

f1 = v / λ1 = sqrt(T / μ) / (2L)

Thus, the formula used by the Fundamental Frequency (f1) Calculator is:

f1 = (1 / 2L) * sqrt(T / μ)

Variables Table

Variable	Meaning	Unit	Typical Range (for a guitar string)
f1	Fundamental Frequency	Hertz (Hz)	80 – 1200 Hz
L	Length of the string	Meters (m)	0.5 – 0.7 m
T	Tension in the string	Newtons (N)	50 – 150 N
μ	Linear mass density	Kilograms per meter (kg/m)	0.0005 – 0.01 kg/m
v	Wave speed	Meters per second (m/s)	100 – 800 m/s
λ1	Fundamental Wavelength	Meters (m)	1.0 – 1.4 m

Practical Examples (Real-World Use Cases)

Example 1: Guitar String

A guitar string has a length of 0.65 m, is under a tension of 80 N, and has a linear mass density of 0.0006 kg/m. Let's use the Fundamental Frequency (f1) Calculator logic:

• L = 0.65 m
• T = 80 N
• μ = 0.0006 kg/m

Wave speed v = sqrt(80 / 0.0006) ≈ sqrt(133333.33) ≈ 365.15 m/s

Fundamental frequency f1 = (1 / (2 * 0.65)) * 365.15 = (1 / 1.3) * 365.15 ≈ 280.88 Hz. This is close to the note D4.

Example 2: Piano Wire

A piano wire for a middle C note might have a length of 0.5 m, tension of 600 N, and linear mass density of 0.002 kg/m.

• L = 0.5 m
• T = 600 N
• μ = 0.002 kg/m

Wave speed v = sqrt(600 / 0.002) = sqrt(300000) ≈ 547.72 m/s

Fundamental frequency f1 = (1 / (2 * 0.5)) * 547.72 = (1 / 1.0) * 547.72 ≈ 547.72 Hz. (This is actually closer to C#5 – piano strings are more complex!). Real piano strings for C4 would be tuned to around 261 Hz using different parameters.

How to Use This Fundamental Frequency (f1) Calculator

1. Enter String Length (L): Input the length of the vibrating part of the string in meters.
2. Enter Tension (T): Input the tension applied to the string in Newtons.
3. Enter Linear Mass Density (μ): Input the mass per unit length of the string in kg/m.
4. View Results: The calculator automatically updates the fundamental frequency (f1), wave speed (v), fundamental wavelength (λ1), and the frequency of the second harmonic (f2) as you type.
5. Reset: Click "Reset" to return to the default values.
6. Interpret Chart & Table: The chart and table visualize the fundamental frequency and the first few overtones (harmonics).

The results from the Fundamental Frequency (f1) Calculator help you understand how changing the physical properties of a string affects the pitch of the sound it produces.

Key Factors That Affect Fundamental Frequency (f1) Results

Several factors influence the fundamental frequency of a vibrating string, as shown by the formula f1 = (1 / 2L) * sqrt(T / μ) and calculated by the Fundamental Frequency (f1) Calculator:

• Length (L): Shorter strings have higher fundamental frequencies (higher pitch). Doubling the length halves the frequency, assuming tension and density remain constant.
• Tension (T): Higher tension results in a higher fundamental frequency (higher pitch). The frequency is proportional to the square root of the tension.
• Linear Mass Density (μ): Strings with lower linear mass density (thinner or lighter strings) have higher fundamental frequencies. The frequency is inversely proportional to the square root of the linear mass density.
• Temperature: Temperature can affect the tension and length of the string slightly, thus indirectly influencing the frequency.
• Material of the String: The material affects the linear mass density (μ) for a given thickness.
• End Conditions: The formula assumes the string is fixed at both ends. Different boundary conditions would result in different modes of vibration and frequencies.

Understanding these factors is crucial when designing or tuning musical instruments that use strings. The Fundamental Frequency (f1) Calculator allows you to explore these relationships.

Frequently Asked Questions (FAQ)

1. What is the difference between fundamental frequency and harmonics?

The fundamental frequency (f1) is the lowest frequency of vibration. Harmonics are frequencies that are integer multiples of the fundamental (f1, 2*f1, 3*f1, …). f1 is the first harmonic.

2. What are overtones?

Overtones are any frequencies produced by an instrument other than the fundamental. For ideal strings and pipes, the overtones are the same as the higher harmonics (2f1, 3f1, etc.).

3. How does the Fundamental Frequency (f1) Calculator handle units?

The calculator expects length in meters (m), tension in Newtons (N), and linear mass density in kg/m. The output frequency is in Hertz (Hz).

4. Can I use this calculator for air columns (like in flutes)?

No, this Fundamental Frequency (f1) Calculator is specifically for stretched strings. The formulas for air columns are different (depending on whether the pipe is open or closed at the ends).

5. Why is the fundamental frequency important in music?

The fundamental frequency largely determines the pitch of the note we hear.

6. What if the string is not uniform?

The formula and this calculator assume a uniform linear mass density. If the string is not uniform, the calculation becomes much more complex.

7. Does the amplitude of vibration affect the frequency?

For small amplitudes, the frequency is largely independent of amplitude. For very large amplitudes, there can be slight changes, but this calculator assumes small amplitudes.

8. What does the chart show?

The chart displays the relative frequencies of the fundamental (f1) and the next three harmonics (f2, f3, f4), giving a visual representation of the harmonic series based on the calculated f1.