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88                                   'ON THE  PBOPAGATION  OF WAVES                                 [
takes place however gradual the passage may be from the first medi to the second, the only condition being that when the second is reac the angle of refraction becomes imaginary. From this point of view t reflection is more naturally regarded as a sort of refraction, reflection pro depending on some degree of abruptness of transition. Phenomena of kind are familiar in Optics under the name of mirage.
In the province of acoustics the vagaries of fog-signals are natur referred to irregular refraction and reflection in the atmosphere, due temperature or wind differences ; but the difficulty of verifying a suggej explanation on these lines is usually serious, owing to our ignorance of state of affairs overhead*.
The penetration of vibrations into a medium where no regular waves be propagated is a matter of considerable interest ; but, so far as I aware, there is no discussion of such a case, beyond that already sketc' relating to a sudden transition between two uniform media. It might 1" been supposed that oblique propagation through a variable medium we involve too many difficulties, but we have already had opportunity to that, in reality, obliquity need not add appreciably to the complicatio] the problem.
To fix ideas, let us suppose that we are dealing with waves in a membi uniformly stretched with tension T, and of superficial density p, which function of x only.    The equation of vibration is (Theory of Sound, § 194)
p dtz or, if $ be proportional to ei(ct+bv], as in (1),
agreeing with (53) if                &2= c-pfT -If- ............................... (9(
The waves originally move towards the less dense parts, and total reflec will ensue when a place is reached, at and after which kn~ is negative. •case which best lends itself to analytical treatment is when p is a li] function of so.    fc2 is then also a linear function ; and, by suitable choic the origin and scale of.*, (95) takes the form
* An observation during the exceptionally hot weather of last summer recalled my atte: to this subject. A train passing at high speed at a distance of 'not more than 150 yardi almost inaudible. The wheels were in full view, but the situation was such that the line of -v passed for most of its length pretty close to the highly heated ground. It seemed clear tha sound rays which should have reached the observers were deflected upwards over their h which were left in a kind of shadow.kness, of which the first is similar to the final uniform medium, and the second similar to the initial uniform medium. •Of the three partial reflections the first and third are similar, but the secondo guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.