applicable, we substitute () into (), obtaining where the undetermined sign is the opposite of that in (). For an evanescent transmitted wave – that is, one whose amplitude decays as increases – the undetermined sign in () must be minus, so the undetermined sign in () must be plus. With the correct sign, the result () can be abbreviated where and is the wavenumber in vacuum, i.e. So the evanescent wave is a plane sinewave traveling in the direction, with an amplitude that decays exponentially in the direction (cf. Fig.9). It is evident that the energy stored in this wave likewise travels in the direction and does not cross the interface. Hence the Poynting vector generally has a component in the direction, but its component averages to zero (although its instantaneous component is not identically zero). Eq.() indicates that the amplitude of the evanescent wave falls off by a factor as the coordinate (measured from the interface) increases by the distance commonly called the "penetration depth" of the evanescent wave. Taking reciprocals of the first equation of (), we find that the penetration depth is where λ0 is the wavelength in vacuum, i.e. Dividing the numerator and denominator by yields where is the wavelength in the second (external) medium. Hence we can plot in units of λ2, as a function of the angle of incidence, for various values of (Fig.12). As θ decreases towards the critical angle, the denominator approaches zero, so that increases without limit – as is to be expected, because as soon as θ is less than critical, uniform plane waves are permitted in the external medium. As θ approaches 90° (grazing incidence), approaches a minimum For incidence from water to air, or common glass to air, is not much different from . But is larger at smaller angles of incidence (Fig.12), and the amplitude may still be significant at distances of several times ; for example, because is just greater than 0.01, the evanescent wave amplitude within a distance of the interface is at least 1% of its value at the interface. Hence, speaking loosely, we tend to say that the evanescent wave amplitude is significant within "a few wavelengths" of the interface. Phase shifts Between 1817 and 1823, Augustin-Jean Fresnel discovered that total internal reflection is accompanied by a non-trivial phase shift (that is, a phase shift that is not restricted to 0° or