Hey, guys, In this video, we're going to delve a little deeper into the mathematical representation of waves and cover a related concept called phase angle. Okay, let's get to it now. We said that a wave that begins with no displacement is a sine wave, right? And a wave that begins with a maximum displacement. A displacement at the amplitude is a cosine wave. But what if we have a wave that begins with a displacement in between the two? Then what happens? Well, in this case, the most complete description of a wave. The best mathematical description of a wave is to combine all of those possibilities in tow one. And we typically right, That is a sign K X minus omega T plus fi, where fi is what we call the phase angle. Okay, the phase angle tells us what that initial displacement is. Okay, the phasing was determined by the initial displacement off the wave. Ah, wave that begins with no displacement has a phase angle of zero degrees, which is a pure sine wave. A wave that begins with a maximum displacement has a phase angle, power to which is a pure cosine wave. Um arbitrary wave One has displacement between zero and its maximum. It's gonna have a phase angle between zero and pi over two. Okay, It's gonna be a mixture of sine and cosine waves. Let's do an example away with a period of seconds and a velocity of 25 m per second has an amplitude of 12 centimeters. At T equals zero and X equals zero. The wave has a displacement of eight centimeters. What is the mathematical representation of this wave? Okay, so we're saying, why is a sign K X minus omega T plus five? Okay, since X and tear both zero, we can ignore them. And all that PFI depends upon is the initial displacement. So this is going to be 12 centimeters sign of five, and that equals eight centimeters. That's the initial displacement. So sign of Phi equals eight centimeters over 12 centimeters. And FYI, which is the inverse sign of 8/12 is simply Okay, very, very quick. Very easy. All right, guys, this wraps up our discussion on the phase angle and the proper representation off a wave as a mathematical function. Thanks for watching