Half wave symmetry and integrals

In summary, having a periodic wave x(t) with half-wave symmetry means that x(t + T0/2) = -x(t) where T0 is the period of the wave. This automatically leads to the conclusion that X(t + T0/2) = -X(t) where X'(t) = x(t), i.e. X(t) is the integral of x(t). This can be proven by integrating both sides of x(t + T0/2) = -x(t) with respect to t.
  • #1
fernlund
10
0
If I have a periodic wave x(t) with half-wave symmetry, it means that:

x(t + T0/2) = -x(t)

where T0 is the period of the wave. Would this automatically lead to the conclusion that

X(t + T0/2) = -X(t)

where X'(t) = x(t), i.e X(t) is the integral of x(t).

?
 
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  • #2
fernlund said:
If I have a periodic wave x(t) with half-wave symmetry, it means that:

x(t + T0/2) = -x(t)

where T0 is the period of the wave. Would this automatically lead to the conclusion that

X(t + T0/2) = -X(t)

where X'(t) = x(t), i.e X(t) is the integral of x(t).

?

I think so. Can you integrate $x(t+T_0/2)=-x(t)$ on both sides w.r.t. $t$, and obtain your result?
 
  • #3
Ackbach said:
I think so. Can you integrate $x(t+T_0/2)=-x(t)$ on both sides w.r.t. $t$, and obtain your result?

Yeah, it's correct for all of the examples I can think of. Thanks!
 

FAQ: Half wave symmetry and integrals

What is half wave symmetry?

Half wave symmetry is a property of a periodic function where the function is symmetrical about the mid-point of one half of its period. This means that the function repeats itself after half of its period, and the values on one half of the period are the same as the values on the other half, but with opposite signs.

How is half wave symmetry related to integrals?

Half wave symmetry is closely related to integrals because it allows us to simplify the calculation of integrals for certain functions. If a function has half wave symmetry, we can integrate over only half of its period and then multiply the result by two to get the total value of the integral.

What types of functions exhibit half wave symmetry?

Functions that are odd and periodic with a period of 2π, such as sine and cosine, exhibit half wave symmetry. This means that any function that can be written in the form f(x) = -f(x + π) will display half wave symmetry.

How can half wave symmetry be used to simplify calculations?

Half wave symmetry can be used to simplify calculations by reducing the number of terms in a function. For example, if we have a function with half wave symmetry, we can rewrite it by only considering the terms that appear in one half of its period, and then multiply the result by two.

Are there any limitations to using half wave symmetry in integrals?

Yes, there are limitations to using half wave symmetry in integrals. It can only be applied to functions that have half wave symmetry, so it cannot be used for all types of integrals. Additionally, it can only be used for periodic functions with a period of 2π, so it is not applicable to all types of functions.

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