Let us monitor the propagation of a travelling wave represented by `y (x, t ) = a sin (kx - ωt + phi )` along a string.
The wave is travelling in the positive direction of x. We find that an element of string at a particular position x moves up and down as a function of time but the waveform advances to the right.
The displacement of various elements of the string at two different instants of time t differing by a small time interval Δt is depicted in Fig.
It is observed that during this interval of time the entire wave pattern moves by a distance Δx in the positive direction of x.
The ratio Δx/Δt is the wave speed v.
Here, `kx - ωt =` constant
To find the wave speed v, let us differentiate it with respect to time
`d/(dt)(kx-omegat)=0`
`k(dx)/(dt)-omega=0`
`(dx)/(dt)=omega/k=v`
`v=omega/k=lamda/T=lamdanu`
`text(Speed of a Transverse Wave on Stretched String :)`
The speed of transverse waves on a string is determined by two factors,
(i) The linear mass density or mass per unit length, `μ`, and
(ii) The tension T.
`v=Csqrt(T/mu)`
Here C is a dimensionless constant that cannot be determined by dimensional analysis. By adopting a more rigorous procedure it can be shown that the constant C is indeed equal to unity. The speed of transverse waves on a stretched string is, therefore, given by
`v=sqrt(T/mu)`
The frequency of the wave is determined by the source that generates the wave.
`lamda=v/nu`
`text(Speed of a Longitudinal Wave Speed of Sound :)`
In a longitudinal wave the constituents of the medium oscillate forward and backward in the direction of propagation of the wave.
The speed of longitudinal waves in a medium is `v=Csqrt(B/rho)`
Where C is a dimensionless constant and can be shown to be unity.
And bulk modulus `B=-(DeltaP)/(DeltaV//V)`
Thus the speed of longitudinal waves in a medium is given by `v=sqrt(B/rho)`
The speed of propagation of a longitudinal wave in a fluid therefore depends only on the bulk modulus and the density of the medium.
When a solid bar is struck a blow at one end, the situation is somewhat different from that of a fluid confined in a tube or cylinder of constant cross section. For this case,
`v=sqrt(Y/rho)`
where Y is the Young's modulus of the material of the bar.
In the case of an ideal gas, the relation between pressure P and volume V is given by
`PV = Nk_BT`
where N is the number of molecules in volume V, `k_B` is the Boltzmann constant and T the temperature of the gas (in Kelvin).
Therefore, for an isothermal change
`VΔP + PΔV = 0`
`=> -(DeltaP)/((DeltaV//V)=P`
`=> B=P`
`v=sqrt(P/rho)`
This relation was first given by Newton and is known as Newton's formula.
Let us monitor the propagation of a travelling wave represented by `y (x, t ) = a sin (kx - ωt + phi )` along a string.
The wave is travelling in the positive direction of x. We find that an element of string at a particular position x moves up and down as a function of time but the waveform advances to the right.
The displacement of various elements of the string at two different instants of time t differing by a small time interval Δt is depicted in Fig.
It is observed that during this interval of time the entire wave pattern moves by a distance Δx in the positive direction of x.
The ratio Δx/Δt is the wave speed v.
Here, `kx - ωt =` constant
To find the wave speed v, let us differentiate it with respect to time
`d/(dt)(kx-omegat)=0`
`k(dx)/(dt)-omega=0`
`(dx)/(dt)=omega/k=v`
`v=omega/k=lamda/T=lamdanu`
`text(Speed of a Transverse Wave on Stretched String :)`
The speed of transverse waves on a string is determined by two factors,
(i) The linear mass density or mass per unit length, `μ`, and
(ii) The tension T.
`v=Csqrt(T/mu)`
Here C is a dimensionless constant that cannot be determined by dimensional analysis. By adopting a more rigorous procedure it can be shown that the constant C is indeed equal to unity. The speed of transverse waves on a stretched string is, therefore, given by
`v=sqrt(T/mu)`
The frequency of the wave is determined by the source that generates the wave.
`lamda=v/nu`
`text(Speed of a Longitudinal Wave Speed of Sound :)`
In a longitudinal wave the constituents of the medium oscillate forward and backward in the direction of propagation of the wave.
The speed of longitudinal waves in a medium is `v=Csqrt(B/rho)`
Where C is a dimensionless constant and can be shown to be unity.
And bulk modulus `B=-(DeltaP)/(DeltaV//V)`
Thus the speed of longitudinal waves in a medium is given by `v=sqrt(B/rho)`
The speed of propagation of a longitudinal wave in a fluid therefore depends only on the bulk modulus and the density of the medium.
When a solid bar is struck a blow at one end, the situation is somewhat different from that of a fluid confined in a tube or cylinder of constant cross section. For this case,
`v=sqrt(Y/rho)`
where Y is the Young's modulus of the material of the bar.
In the case of an ideal gas, the relation between pressure P and volume V is given by
`PV = Nk_BT`
where N is the number of molecules in volume V, `k_B` is the Boltzmann constant and T the temperature of the gas (in Kelvin).
Therefore, for an isothermal change
`VΔP + PΔV = 0`
`=> -(DeltaP)/((DeltaV//V)=P`
`=> B=P`
`v=sqrt(P/rho)`
This relation was first given by Newton and is known as Newton's formula.