Exploring Complex Wave Equations: From (8) to (9)

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The transition from equation (8) to equation (9) involves expressing wave equations in complex form, even without an explicit imaginary part like isin(omega*t-kz+phi). This approach is largely a matter of convention, as the imaginary component is often disregarded for simplicity. Using complex exponentials facilitates easier mathematical operations such as addition and multiplication compared to traditional trigonometric forms. While this method is common, it is important to note that in fields like Quantum Mechanics, the imaginary part can be crucial. Overall, the use of complex forms streamlines calculations in wave physics.
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From equation (8) to equation (9) fro the link attached below which the equation is written into a complex form. But why can it be written in that form even though the equation do not have an imaginary part isin(omega*t-kz+phi)?

http://scienceworld.wolfram.com/physics/WaveEquation.html
 
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Good question :smile:

It's a matter of convention, we usually just ignore the imaginary part (note that this is not always the case though, for instance in Quantum Mechanics we often need the imaginary part as well). The reason waves are usually expressed as complex exponentials is that it makes the math easier (believe it or not), it's a lot easier to perform various additions, multiplications, etc. with an exponential than it is to try and slough through trig identities.
 

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