diff --git a/tornado/websocket.py b/tornado/websocket.py
index fbfd700887..fc7bf1b30f 100644
--- a/tornado/websocket.py
+++ b/tornado/websocket.py
@@ -11,6 +11,8 @@
    Removed support for the draft 76 protocol version.
 """
 
+from time import time
+
 import abc
 import asyncio
 import base64
@@ -835,7 +837,6 @@ def __init__(
         self._wire_bytes_in = 0
         self._wire_bytes_out = 0
         self.ping_callback = None  # type: Optional[PeriodicCallback]
-        self.last_ping = 0.0
         self.last_pong = 0.0
         self.close_code = None  # type: Optional[int]
         self.close_reason = None  # type: Optional[str]
@@ -1230,6 +1231,7 @@ def _handle_message(self, opcode: int, data: bytes) -> "Optional[Future[None]]":
             self.close(self.close_code)
         elif opcode == 0x9:
             # Ping
+            print(f':ping: {time()}')
             try:
                 self._write_frame(True, 0xA, data)
             except StreamClosedError:
@@ -1237,6 +1239,7 @@ def _handle_message(self, opcode: int, data: bytes) -> "Optional[Future[None]]":
             self._run_callback(self.handler.on_ping, data)
         elif opcode == 0xA:
             # Pong
+            print(f':pong: {time()}')
             self.last_pong = IOLoop.current().time()
             return self._run_callback(self.handler.on_pong, data)
         else:
@@ -1303,38 +1306,34 @@ def start_pinging(self) -> None:
         """Start sending periodic pings to keep the connection alive"""
         assert self.ping_interval is not None
         if self.ping_interval > 0:
-            self.last_ping = self.last_pong = IOLoop.current().time()
+            if self.ping_timeout and self.ping_timeout > self.ping_interval:
+                raise ValueError(
+                    'the ping_timeout cannot be greater than the ping_interval'
+                )
             self.ping_callback = PeriodicCallback(
                 self.periodic_ping, self.ping_interval * 1000
             )
             self.ping_callback.start()
 
-    def periodic_ping(self) -> None:
-        """Send a ping to keep the websocket alive
+    async def periodic_ping(self) -> None:
+        """Send a ping and wait for a pong if ping_timeout is configured.
 
         Called periodically if the websocket_ping_interval is set and non-zero.
         """
-        if self.is_closing() and self.ping_callback is not None:
-            self.ping_callback.stop()
-            return
-
-        # Check for timeout on pong. Make sure that we really have
-        # sent a recent ping in case the machine with both server and
-        # client has been suspended since the last ping.
         now = IOLoop.current().time()
-        since_last_pong = now - self.last_pong
-        since_last_ping = now - self.last_ping
-        assert self.ping_interval is not None
-        assert self.ping_timeout is not None
-        if (
-            since_last_ping < 2 * self.ping_interval
-            and since_last_pong > self.ping_timeout
-        ):
-            self.close()
-            return
 
+        # send a ping
+        print(f':ping: {time()}')
         self.write_ping(b"")
-        self.last_ping = now
+
+        if self.ping_timeout and self.ping_timeout > 0:
+            # wait for the pong
+            await asyncio.sleep(self.ping_timeout)
+
+            # close the connection if the pong is not received within the
+            # configured timeout
+            if self.last_pong - now > self.ping_timeout:
+                self.close()
 
     def set_nodelay(self, x: bool) -> None:
         self.stream.set_nodelay(x)