A Surface Built from Repeating Waves
The function \(z=\sin x+\sin y\) combines two familiar waves: one varies as \(x\) changes, and the other as \(y\) changes. Its graph is a surface above the \(xy\)-plane. Because each sine term repeats, the surface repeats in both horizontal directions. The result is a regular pattern of ridges, valleys, and contours rather than a single isolated peak or dip.
Angles in this tutorial are measured in radians. To picture the surface, it helps to look at it in two complementary ways. A coordinate trace fixes one input and shows the resulting curve in a vertical plane. A level curve fixes the height \(z=c\) and shows the corresponding points in the \(xy\)-plane. These are the same kinds of views used in earlier material on traces and level curves, now applied to a periodic surface.
Coordinate Traces Show the Repeated Ripples
Fixing \(y=b\) leaves \(x\) as the changing input. The trace in the vertical plane \(y=b\) is therefore \(z=\sin x+\sin b\): a sine wave shifted vertically by the constant \(\sin b\). Fixing \(x=a\) gives the corresponding trace \(z=\sin a+\sin y\) in the plane \(x=a\). Each trace oscillates between a minimum and a maximum, but its vertical shift depends on the fixed coordinate.
Worked Example: A Trace at \(y=\frac{\pi}{6}\)
Substitute \(y=\frac{\pi}{6}\) into the surface equation. Since \(\sin(\frac{\pi}{6})=\frac12\), the trace is
Evaluate this trace at four inputs. At \(x=0\), \(z=0+\frac12=\frac12\). At \(x=\frac{\pi}{2}\), \(z=1+\frac12=\frac32\). At \(x=\frac{3\pi}{2}\), \(z=-1+\frac12=-\frac12\). At \(x=2\pi\), \(z=0+\frac12=\frac12\).
Thus the trace passes through \((0,\frac12)\), \((\frac{\pi}{2},\frac32)\), \((\frac{3\pi}{2},-\frac12)\), and \((2\pi,\frac12)\) in its \(xz\)-plane. The final height matches the first because the sine wave has completed one period. Substitution into \(z=\sin x+\frac12\) verifies each height.
In the full surface, the fixed-\(y\) traces have the same wave shape but different vertical positions. For instance, at \(y=\frac{\pi}{2}\), the trace is \(z=\sin x+1\), while at \(y=\frac{3\pi}{2}\), it is \(z=\sin x-1\). Looking at a few such traces helps show how the surface rises or falls as \(y\) changes.
Worked Example: Comparing Two Traces
First fix \(y=\frac{\pi}{2}\). Since \(\sin(\frac{\pi}{2})=1\), the trace is \(z=\sin x+1\). At \(x=0\), its height is \(1\); at \(x=\frac{\pi}{2}\), its height is \(2\); and at \(x=\frac{3\pi}{2}\), its height is \(0\).
Now fix \(y=\frac{3\pi}{2}\). Since \(\sin(\frac{3\pi}{2})=-1\), the trace is \(z=\sin x-1\). At the same three \(x\)-values, the heights are \(-1\), \(0\), and \(-2\). Each value checks directly: for example, at \((x,y)=(\frac{\pi}{2},\frac{3\pi}{2})\), \(z=1+(-1)=0\).
Both traces have the same wave shape, but the second lies two units below the first. The shift comes from changing the fixed \(y\)-value, not from changing the period of the wave.
Height Limits and Repeating Surface Features
For every real input, \(-1\leq\sin x\leq1\) and \(-1\leq\sin y\leq1\). Adding these inequalities gives \(-2\leq\sin x+\sin y\leq2\). Both bounds are attained: the upper bound requires both sine terms to equal \(1\), and the lower bound requires both to equal \(-1\).
The sine function repeats after a shift of \(2\pi\): \(\sin(t+2\pi m)=\sin t\) for every integer \(m\). Applying this fact independently to \(x\) and \(y\) shows why the whole surface repeats across the plane.
In particular, translating a point in the \(xy\)-plane by \(2\pi\) in either coordinate leaves its height unchanged. A peak, valley, or contour feature therefore has copies at all such translated locations. The repeated pattern is a direct consequence of the formula, not just a visual impression from a sketch.
Worked Example: Locating Repeated Highest Points
For the height to equal \(2\), each sine term must equal \(1\). The solutions are \(x=\frac{\pi}{2}+2\pi m\) and \(y=\frac{\pi}{2}+2\pi n\), where \(m,n\in\mathbb Z\). Thus the highest points on the graph have coordinates
For example, choosing \(m=n=0\) gives \((\frac{\pi}{2},\frac{\pi}{2},2)\), where \(\sin(\frac{\pi}{2})+\sin(\frac{\pi}{2})=1+1=2\). Choosing \(m=1\) and \(n=0\) gives \((\frac{5\pi}{2},\frac{\pi}{2},2)\); its height is also \(1+1=2\), because \(\sin(\frac{5\pi}{2})=\sin(\frac{\pi}{2})=1\). These substitutions verify both the maximum height and its repetition.
Level Curves Reveal the Pattern from Above
At a fixed height \(c\), the level curve equation is \(\sin x+\sin y=c\). If \(|c|>2\), there are no points on that level, because the sum cannot be higher than \(2\) or lower than \(-2\). At \(c=2\) or \(c=-2\), only the repeated highest or lowest points in the plane satisfy the equation. For intermediate heights, the level curves show how combinations of the two waves produce the same total height.
One especially useful contour is \(c=0\). The identity \(\sin x+\sin y=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})\) factors its equation into two simple conditions. A product is zero exactly when at least one factor is zero; solving each condition gives two families of lines.
Worked Example: Deriving and Checking the Zero Contour
Set \(c=0\), so \(\sin x+\sin y=0\). Apply the sum-to-product identity:
The first factor is zero when \(\frac{x+y}{2}=k\pi\), or \(x+y=2k\pi\). The second is zero when \(\frac{x-y}{2}=\frac{\pi}{2}+k\pi\), or \(x-y=(2k+1)\pi\). Together, these are the two stated families.
Check a point from each family. The point \((0,0)\) satisfies \(x+y=0\), and \(\sin 0+\sin 0=0\). The point \((0,\pi)\) satisfies \(x-y=-\pi\), which has the form \((2k+1)\pi\) for \(k=-1\); its height is \(\sin 0+\sin\pi=0+0=0\). Both points lie on the zero contour, as required.
Because the level equation is unchanged by replacing \(x\) with \(x+2\pi m\) and \(y\) with \(y+2\pi n\), every level curve repeats under the same translations. On a contour map, this means that a patch of contours repeats every \(2\pi\) horizontally and vertically. A contour’s shape can be more complicated than a coordinate trace, but the periodic copies remain exact.
A Short Justification of the Repeated Contours
Let \((x,y)\) be any point on the level curve at height \(c\). By definition, \(\sin x+\sin y=c\). Translate the point to \((x+2\pi m,y+2\pi n)\), where \(m,n\) are integers. Periodicity gives \(\sin(x+2\pi m)=\sin x\) and \(\sin(y+2\pi n)=\sin y\). Therefore the translated point also satisfies the same level equation:
This proves that every translated copy of a point on a level curve remains on that level curve. Applying the same reasoning with the shifts reversed shows that the repeated copies cover the pattern in both directions. This is why a contour map of the surface shows recurring arrangements rather than unrelated shapes in different regions.
A Common Height Mistake
A common error is to assume that the maximum height is \(1\), because the maximum value of each sine function is \(1\). That overlooks the addition in the formula. At \(x=\frac{\pi}{2}\) and \(y=\frac{\pi}{2}\), the actual calculation is
Likewise, at \(x=\frac{3\pi}{2}\) and \(y=\frac{3\pi}{2}\), the height is \(-1+(-1)=-2\). The bound for the sum must account for both terms. A reliable sketch labels the heights of the combined surface, not just the range of either individual sine wave.
Using the Views Together
Coordinate traces and level curves emphasize different information. A trace records height as one coordinate varies while the other is held fixed. A contour records which pairs of coordinates produce one chosen height. For this surface, traces make the wave pattern and vertical shifts easy to calculate; contours show how the two periodic waves combine across the plane.
To sketch a useful local patch, mark a few familiar sine values along each direction, calculate heights by addition, and note where the maximum and minimum heights occur. Then use the contour equation to identify selected level sets and repeat the resulting features every \(2\pi\) in either coordinate. The aim is not to draw every contour exactly, but to preserve the height, shape, and repetition implied by the formula.
Check Your Understanding
Use the trace and level-curve ideas to answer these questions.
- What is the trace equation when \(x=\pi\), and what is its height at \(y=\frac{\pi}{2}\)?
- What is the largest possible height of \(z=\sin x+\sin y\), and which sine values produce it?
- Explain why replacing \(y\) by \(y+2\pi\) leaves the surface height unchanged.
- What two families of lines make up the level curve at height \(0\)?
- Why is a surface height of \(3\) impossible for this function?