Transverse and longitudinal standing waves are easily represented by sine waves. Positions of maximum displacement from equilibrium are called antinodes A, positions of zero displacement are called nodes, N.

For a complete cycle of a wave of wavelength λ the distance between any two consecutive nodes (N - A - N) or any two consecutive antinodes (A - N - A) is half of the wavelength, . The distance between an adjacent node and antinode (N - A) is a quarter of the wavelength, .

Longitudinal standing waves can be produced in columns of air in pipes. The closed end of a pipe is always a node. The open end of a pipe is always an antinode.

Wavelength, λ, and frequency, µ, of mechanical waves are related to the speed of the wave through the given medium : λµ = v

For room temperature air, the speed of sound, v, is 340 m/s.

The lowest natural frequency is called the fundamental frequency, µ 0 , and has wavelength λ 0 . A harmonic is an integral multiple of the fundamental frequency, nµ0, with n = 1, 2, 3, ...... So the fundamental frequency is the first harmonic. The harmonics, that occur naturally are called overtones.

(I) Fundamental and first overtone for a pipe open at both ends.

= L

Therefore λ 0 = 2L and v 0 =

Therefore,λ 1 = 2 and v 1 =

Overtones are multiples of v 0

For the first overtone n = 2, the second

nµ0 = harmonic. This implies that the ratio of natural frequencies is n = 1 : 2 : 3 : .......

(II) Fundamental and first overtone for a pipe closed at one end

= L,

Therefore λ 0 = 4L and µ 0 = Therefore,

λ 1 = 4 and

Overtones are multiples of µ 0 :

n µ 0 =

For the first overtone n = 3, the third harmonic. This implies that the ratio of natural frequencies is n = 1 : 3 : 5 : ...

(i) The auditory canal of the outer ear is essentially a 3.0 cm pipe closed at one end by the eardrum. What is the fundamental frequency in air associated with the ear?