# Phase, Group, and Signal Velocity

Since a general wave (or wavelike phenomenon) need not embody the causal flow of any physical effects, there is obviously there is no upper limit on the possible phase velocity of a wave. However, even for a “genuine” physical wave, i.e., a chain of sequentially dependent events, the phase velocity does not necessarily correspond to the speed at which energy or information is propagating. This is partly a semantical issue, because in order to actually convey information, a signal cannot be a simple periodic wave, so we must consider non-periodic signals, making the notion of “phase” somewhat ambiguous. If the wave profile never exactly repeats itself, then arguably the “period” of the signal must be the entire signal. On this basis we might say that the velocity of the signal is unambiguously equal to the “phase velocity”, but in this context the phase velocity could only be defined as the speed of the leading (or trailing) edge of the overall signal.

— Phase, Group, and Signal Velocity

— mathpages

2013.07.02 Tuesday ACHK

# Apology

Punishment 3 | 以直報怨 3 | Regret 3

sincere apology

~ declaration of change of identity

~ declaration of not inheriting your previous evil self(s)

~ declaration of cutting the evil causal chain

— Me@2013-06-30 6:13 PM

# Multinomial coefficient 2.2

（CYW：但是，我覺得應該不止有 210 個可能性，因為抽某 4 個人出來時，本身有很多個抽法。假設「甲、乙、丙、丁」四人被抽中，去乘坐第一輛車，「先抽甲出來」和「先抽乙出來」，就已經是兩個不同的可能性。我不太明白，為什麼毋須考慮這一點？）

（CYW：用你這個講法，我好像明白多了一點。但是，如果沿用我剛才的問法，我又確實感覺到，應該考慮「抽哪 4 個人去第一輛車」的不同次序。我暫時還未有信心，可以在考試時準確分辨，哪些時候需要考慮「次序」，哪些時候不需要。）

— Me@2013.07.01