APPENDIX.
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Note A ([see p. 21]).
The distinction between the individual wave-velocity and a wave-group velocity, to which, as stated in the text, attention was first called by Sir G. G. Stokes in an Examination question set at Cambridge in 1876, is closely connected with the phenomena of beats in music.
If two infinitely long sets of deep-sea waves, having slightly different wave-lengths, and therefore slightly different velocities, are superimposed, we obtain a resultant wave-train which exhibits a variation in wave-amplitude along its course periodically. If we were to look along the train, we should see the wave-amplitude at intervals waxing to a maximum and then waning again to nothing. These points of maximum amplitude regularly arranged in space constitute, as it were, waves on waves. They are spaced at equal distances, and separated by intervals of more or less waveless or smooth water. These maximum points move forward with a uniform velocity, which we may call the velocity of the wave-train, and the distance between maximum and maximum surface-disturbances may be called the wave-train length.
Let v and v′ be the velocities, and n and n′ the frequencies, of the two constituent wave-motions. Let λ and λ′ be the corresponding wave-lengths. Let V be the wave-train velocity, N the wave-train frequency, and L the wave-train length. Then N is the number of times per second which a place of maximum wave-amplitude passes a given fixed point.
Then we have the following obvious relations:—
v = nλ, v′ = n′λ′, N = n – n′ = v / λ – v′ / λ′
Also a little consideration will show that—
L / λ′ = λ / λ – λ′
since λ is nearly equal, by assumption, to λ′. Hence we have—
1 / L = 1 / λ – 1 / λ′ ; and also V = NL
Accordingly—
V = / N / 1 / L = v / λ – v′ / λ′ / 1 / λ – 1 / λ′
Let us write 2π / k instead of λ, and 2π / k′ instead of λ′; then we have—
V = vk – v′k′ / k – k′ (i.)
And since k and k′, v and v′ are nearly equal, we may write the above expression as a differential coefficient; thus—
V = d(vk) / d(k) (ii.)
Suppose, then, that, as in the case of deep-sea waves, the wave-velocity varies as the square root of the wave-length. Then if C is a constant, which in the case of gravitation waves is equal to g/2π , where g is the acceleration due to gravity, we have—
v² = Cλ, or v² = g / 2π λ
But λ = 2π / k , hence—
vk = 2πC / v
Hence if we differentiate with respect to v, we have—
d(vk) / dv = – 2πC / v²
Again, k = 2π / λ = 2πC / v² ; therefore—
d(k) / dv = – 2 2πC / v³
Hence, dividing the expression for d(vk) / dv by that for d(k) / dv , we have—
V = d(vk) / d(k) = v / 2
In other words, the wave-train velocity is equal to half the wave-velocity. This is the case with deep-sea waves. Suppose, however, that, as in the case of air waves, the wave-velocity is independent of the wave-length. Then if two trains of waves of slightly different wave-length are superposed, we have k and k′ different in value but nearly equal, and v and v′ equal. Hence the equation (i.) takes the form—
V = v
In other words, the beats travel forward with the same speed as the constituent waves. And in this case there is no difference between the velocity of the wave-train and the velocity of the individual wave. The above proof may be generalized as follows:—
Let the wave-velocity vary as the nth root of the wave-length, or let vⁿ = Cλ; and let λ = 2π / k as before.
Then—
vⁿ = 2πC / k , and vk = 2πC / v ⁿ ⁻¹ = 2πCv ⁻ ⁽ ⁿ ⁻¹⁾
also k = 2π / λ = 2πC / v ⁿ = 2πCv ⁻ ⁿ
Hence d(vk) / d(k) = n – 1v ⁻ ⁽ⁿ ⁻¹⁾ ⁻¹ / nv ⁻ ⁿ ⁻¹ = n – 1 / n v
or V = n – 1 / n v
That is, the wave-train velocity is equal to n – 1 / n times the wave-velocity.
In the case of sea waves n = 2, and in the case of air waves n = infinity.
If n were 3, then V = 2 / 3 v, or the group-velocity would be two-thirds the wave-velocity.
Note B ([see p. 273]).
Every electric circuit comprising a coil of wire and a condenser has a definite time-period in which an electric charge given to it will oscillate if a state of electric strain in it is suddenly released. Thus the Leyden jar L and associated coil P shown in [Fig. 82, p. 271], constitutes an electric circuit, having a certain capacity measured in units, called a microfarad, and a certain inductance, or electric inertia measured in centimetres. The capacity of the circuit is the quality of it in virtue of which an electric strain or displacement can be made by an electromotive force acting on it. The inductance is the inertia quality of the circuit, in virtue of which an electric current created in it tends to persist. In the case of mechanical oscillations such as those made by vibrating a pendulum, the time of one complete oscillation, T, is connected with the moment of inertia, I, and the mechanical force brought into play by a small displacement as follows: Suppose we give the pendulum a small angular displacement, denoted by θ. Then this displacement brings into existence a restoring force or torque which brings the pendulum back, when released, to its original position of rest. In the case of a simple pendulum consisting of a small ball attached to a string, the restoring torque created by displacing the pendulum through a small angle, θ, is equal to the product mglθ, where m is the mass of the bob, g is the acceleration of gravity, and l is the length of the string. The ratio of displacement (θ) to the restoring torque mglθ is 1 / mgl . This may be called the displacement per unit torque, and may otherwise be called the pliability of the system, and denoted generally by P. Let I denote the moment of inertia. This quantity, in the case of a simple pendulum, is the product of the mass of the bob and the square of the length of the string, or I = ml².
In the case of a body of any shape which can vibrate round any centre or axis, the moment of inertia round this axis of rotation is the sum of the products of each element of its mass and the square of their respective distances from this axis. The periodic time T of any small vibration of this body is then obtained by the following rule:—
| T = 2π | √ | moment of inertia round | } × { | displacement per unit of |
| the axis of rotation | torque, or pliability |
or T = 2π√IP.
In the case of an electric circuit the inductance corresponds to the moment of inertia of a body in mechanical vibration; and the capacity to its pliability as above defined. Hence the time of vibration, or the electrical time-period of an electric circuit, is given by the equation—
T = 2π√LC
where L is the inductance, and C is the capacity.
It can be shown easily that the frequency n, or number of electrical vibrations per second, is given by the rule—
| n = | 5000000 | |||
| √ | capacity in | } × { | inductance in | |
| microfarads | centimetres | |||
For instance, if we discharge a Leyden jar having a capacity of ¹⁄₃₀₀ of a microfarad through a stout piece of copper wire about 4 feet in length and one-sixth of an inch in diameter, having an inductance of about 1200 centimetres, the electrical oscillations ensuing would be at the rate of 2¹⁄₂ millions per second.
Any two electrical circuits which have the same time-period are said to be “in tune” with each other, and the process of adjusting the inductance and capacity of the circuits to bring about this result is called electrical tuning. In the case of a vertical aerial wire as used in wireless telegraphy, in which the oscillations are created by the inductive action of an oscillation-transformer as shown in [Fig. 82, page 271], the capacity of the Leyden jar in the condenser circuit must be adjusted so that the time-period of the nearly closed or primary oscillation P agrees with that of the open or secondary circuit S. When this is the case, the electrical oscillations set up in the closed circuit have a far greater effect in producing others in the open circuit than if the two circuits were not in tune. The length of the wave given off from the open circuit is approximately equal to four times the length of the aerial wire, including the length of the coil forming the secondary circuit of the oscillation-transformer in series with it.