The essence and importance of linear phase property lies in the definition and the effect of **group delay** $$\tau(\omega) = - \frac {d\phi(\omega)}{d\omega}$$ on the applied signal $x[n]$, where $\phi(\omega)$ is the **phase** response of the filter; (phase of its frequency response). Assume that a filter, with a fixed group delay of $n_0$ samples, is applied a **narrowband** input signal $x[n]$. Then the output signal will be (approximately) of the form $y[n] = K x[n-n_0]$ where $K$ is the filter gain evaluated at the center frequency of the narrowband input signal $x[n]$. This means that the input signal will be weighted and shifted **intact** as a whole by the group delay of the filter. And this can only happen when the group delay is **independent** of the frequency $\omega$. And this will be the case if the underlying filter has **linear phase** (or generalized linear phase). Note that if the input signal is of broadband type; i.e., its minimum and maximum frequencies are far from its center frequency, then the approximation is not valid and eventhough the group delay would still be the same for each sinusoidal component in the signal, their relative output amplitudes will differ by the frequency dependent filter gain $K(w)$. Then what's the effect of a filter with nonlinear phase (or frequency dependent group delay) on the input signal? A simple example would be a complicated input signal considered as a sum of multiple wavepackets at different center frequencies. After the filtering, each packet with a particular center frequency will be **shifted** (delayed) differently due to frequency dependent group delay. And this will be resulting in a change in the time-order (or space order) of those wave packets, sometimes drastically, depending on how nonlinear the phase is, which is called as **dispersion** in communications terminalology. Not only the composite waveshape, but also some event orders may be lost. This kind of dispersive channels have severe effects such as ISI (inter symbol interference) on transmitted data. This property of linear phase filters, therefore, is also known as **waveform-preserving** property, which is applicable to narrowband signals in particular. An example where waveform is important, other than ISI as mentioned above, is in processing of images, where the Fourier transform **phase** information is of paramount importance compared to magnitude of Fourier transform, for intelligibility of the image. The same, however, cannot be said for perception of **sound** signals due to a different kind of sensitivity of the ear to the stimulus.