Standing Waves and Harmonics on a String

Introduction to Standing Waves

When a string is fixed at both ends and vibrated, standing waves can form at specific frequencies called harmonics. The properties of these waves depend on the string's length, tension, and mass per unit length.

• Standing Wave: A wave that remains in a constant position, characterized by nodes (points of no motion) and antinodes (points of maximum motion).

• Harmonic: A frequency at which standing waves are established. The first harmonic is the fundamental frequency; higher harmonics are integer multiples of this frequency.

Wave Properties and Equations

• Length of String (L): The distance between the two fixed ends. In this example, .

• Tension (F_T): The force stretching the string, here .

• Frequency (f): The number of oscillations per second. The second harmonic frequency is given as .

• Mass of String (m): The total mass, which is to be determined.

Harmonics and Wavelengths

• For a string fixed at both ends, the th harmonic has antinodes and nodes.

• The wavelength for the th harmonic:

• For the second harmonic ():

Wave Speed and Frequency Relationship

• Wave speed (v):

• For the second harmonic:

Wave Speed and String Properties

• The speed of a wave on a stretched string is given by: where is the mass per unit length ().

• Rearranging for :

Calculation Example

• Given: , ,

• Calculate :

Summary Table: Key Relationships

Example Application

• Musical instruments such as guitars and violins use standing waves on strings to produce sound. The pitch depends on the string's length, tension, and mass per unit length.

Additional info: The notes and calculations shown are typical for introductory physics problems involving waves on strings, harmonics, and the relationship between frequency, tension, and mass.