In: Physics
In terms of λ, f, μ, and T what times the square root of the tension is equal to the wavelength?
Please show work, thanks.
The speed of propagation of a wave in a string () is proportional to the square root of the
tension of the string (
) and inversely proportional to the square
root of the linear density (
) of the string:
Derivation:
Let be the length of a piece of string,
its mass, and
its linear density. If the horizontal
component of tension in the string is a constant,
, then the tension acting on each side of the
string segment is given by
If both angles are small, then the tensions on either side are
equal and the net horizontal force is zero. From Newton's second
law for the vertical component, the mass of this piece times its
acceleration, , will be equal to the net force on the
piece:
Dividing this expression by and substituting the first and second
equations obtains
The tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of beta. Using this fact and rearranging provides
In the limit that approaches zero, the left hand side is the
definition of the second derivative of
:
This is the wave equation for , and the coefficient of the second time
derivative term is equal to
; thus
where is the speed of propagation of the wave in
the string. (See the article on the wave equation for more about
this). However, this derivation is only valid for vibrations of
small amplitude; for those of large amplitude,
is not a good approximation for the length
of the string piece, the horizontal component of tension is not
necessarily constant, and the horizontal tensions are not well
approximated by
.
Once the speed of propagation is known, the frequency of the
sound produced by the string can be calculated. The speed of
propagation of a wave is equal to the wavelength divided by the period
, or multiplied by the frequency
:
If the length of the string is , the fundamental harmonic is the one
produced by the vibration whose nodes are the two ends of the
string, so
is half of the wavelength of the fundamental
harmonic. Hence one obtains Mersenne's laws:
where is the tension (in Newton),
is the linear density (that is, the mass per
unit length), and
is the length of the vibrating part of the
string.
lambda=(1/f) sqrt (T/mu)
This is the required derivation.